Sparsity and dimension - Semantic Scholar
SPARSITY AND DIMENSION
arXiv:1507.01120v1 [math.CO] 4 Jul 2015
¨ JORET, PIOTR MICEK, AND VEIT WIECHERT GWENAEL
Abstract. We prove that posets of bounded height whose cover graphs belong to a fixed class with bounded expansion have bounded dimension. Bounded expansion, introduced by Neˇsetˇril and Ossona de Mendez as a model for sparsity in graphs, is a property that is naturally satisfied by a wide range of graph classes, from graph structure theory (graphs excluding a minor or a topological minor) to graph drawing (e.g. graphs with constant book thickness). Therefore, our theorem generalizes a number of results including the most recent one for posets of bounded height with cover graphs excluding a fixed graph as a topological minor (Walczak, SODA 2015). We also show that the result is in a sense best possible, as it does not extend to nowhere dense classes; in fact, it already fails for cover graphs with locally bounded treewidth.
´ Libre de Bruxelles, Brussels, (G. Joret) Computer Science Department, Universite Belgium (P. Micek) Theoretical Computer Science Department, Faculty of Mathematics and ´ w, Poland and Institut fu ¨ r MatheComputer Science, Jagiellonian University, Krako ¨ t Berlin, Berlin, Germany matik, Technische Universita ¨ r Mathematik, Technische Universita ¨ t Berlin, Berlin, (V. Wiechert) Institut fu Germany E-mail addresses: [email protected], [email protected], [email protected]. Date: July 7, 2015. 2010 Mathematics Subject Classification. 06A07, 05C35. Key words and phrases. Bounded expansion, poset, dimension, cover graph, graph minor. V. Wiechert is supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK 1408). 1
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1. Introduction 1.1. Poset dimension and cover graphs. Partially ordered sets, posets for short, are studied extensively in combinatorics, set theory and theoretical computer science. One of the most important measures of complexity of a poset is its dimension. The dimension dim(P ) of a poset P is the least integer d such that points of P can be embedded into Rd in such a way that x < y in P if and only if the point of x is below the point of y with respect to the product order on Rd . Equivalently, the dimension of P is the least d such that there are d linear extensions of P whose intersection is P . Not surprisingly, dimension is hard to compute: Already deciding whether dim(P ) 6 3 is an NP-hard problem [25], and for every ε > 0, there is no n1−ε -approximation algorithm for dimension unless ZPP=NP, where n denotes the size of the input poset [2]. Research in this area typically focus on finding witnesses for large dimension and sufficient conditions for small dimension; see e.g. [22] for a survey. Posets are visualized by their diagrams: Points are placed in the plane and whenever a < b in the poset, and there is no point c with a < c < b, there is a curve from a to b going upwards (that is y-monotone). The diagram represents those relations which are essential in the sense that they are not implied by transitivity, known as cover relations. The undirected graph implicitly defined by such a diagram is the cover graph of the poset. That graph can be thought of as encoding the ‘topology’ of the poset. There is a common belief that posets having a nice or well-structured drawing should have small dimension. But first let us notea negative observation by Kelly [12], from 1981, there is a family of posets whose diagrams can be drawn without edge crossings— undoubtedly qualifying as a ‘nice’ drawing—and with arbitrarily large dimension, see Figure 1. A key observation about Kelly’s construction is that these posets also have large height. This leads us to our main theorem, which generalizes several previous works in this area. Theorem 1. For every class of graphs C with bounded expansion, and for every integer h > 1, posets of height h whose cover graphs are in C have bounded dimension. 1.2. Background. Theorem 1 takes its roots in the following result of Streib and Trotter [21]: Posets with planar cover graphs have dimension bounded in terms of their height. Note that this justifies the fact that height and dimension grow together in Kelly’s construction. Joret, Micek, Milans, Trotter, Walczak, and Wang [10] subsequently proved that the same result holds in the case of cover graphs of bounded treewidth, of bounded genus, and more generally cover graphs that forbid a fixed apex graph as a minor. In another direction, F¨ uredi and Kahn [9] showed that posets with cover graphs of bounded maximum degree have dimension bounded in terms of their height.1 All this was recently generalized by Walczak [24]: Posets whose cover graphs 1We
note that the original statement of F¨ uredi and Kahn’s theorem is that posets with comparability graphs of bounded maximum degree have bounded dimension; in fact, they show a O(∆ log2 ∆) upper bound on the dimension, where ∆ denotes the maximum degree. Observe however that the comparability graph of a poset has bounded maximum degree if and only its cover graph does and its height is bounded.
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b4 b1
b2
b3
b4
a1
a2
a3
a4
ai < bj iff i 6= j
b3 b2 b1
a1 a2 a3 a4
Figure 1. Standard example S4 (left) and Kelly’s construction of a planar poset containing S4 (right). Recall that the standard example Sd is the poset on 2d points consisting of d minimal points a1 , . . . , ad and d maximal points b1 , . . . , bd such that ai < bj in Sd if and only if i 6= j. It is well-known and easily verified that dim(Sd ) = d. For every d > 1, Kelly’s construction provides a planar poset with 4d − 2 points containing Sd as a subposet, and hence having dimension at least d; its general definition is implicit in the figure. exclude a fixed graph as a topological minor have dimension bounded by a function of their height. (We note that Walczak’s original proof relies on the graph structure theorems for graphs excluding a fixed topological minor; see [14] for an elementary proof.) Bounded degree graphs, planar graphs, bounded treewidth graphs, and more generally graphs avoiding a fixed (topological) minor are sparse, in the sense that they have linearly many edges. This remark naturally leads one to ponder whether simply having a sparse cover graph is enough to guarantee the dimension be bounded by a function of the height. This is exactly the question we address in this work. Our main contribution is to characterize precisely the type of sparsity that is needed to ensure the property. Before explaining it let us make some preliminary observations. Clearly, asking literally that the cover graph has at most cn edges for some constant c, where n is the number of points of the poset,√is not enough since one could simply consider the union of a standard example on Θ( n) points and a large antichain. Slightly less trivially, even requiring that property to hold for every subgraph of the cover graph (that is, bounding its degeneracy) is still not enough: Define the incidence poset IG of a graph G as the height-2 poset with point set V (G) ∪ E(G), where for a vertex v and an edge e we have v < e in PG whenever v is an endpoint of e in G (and no other relation). The fact that dim(IKn ) > log log n follows easily from repeated applications of the Erd˝os-Szekeres theorem for monotone sequences (see for instance [5]). Yet, the cover graph of IKn is a clique where each edge is subdivided once, which is 2-degenerate. 1.3. Nowhere dense classes and classes with bounded expansion. Neˇsetˇril and Ossona de Mendez [17] carried out a thorough study of sparse classes of graphs over the
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last decade. Most notably, they introduced the notions of nowhere dense classes and classes with bounded expansion. Let us give some informal intuition, see Section 2 for the precise definitions. The key idea in both cases is to look at minors of bounded depth,
Bounded average degree Nowhere dense
Bounded degeneracy
Bounded expansion Almost wide
Theorem 1
Bounded queue number
Example 2
Bounded book thickness Excluded topological minor
Locally bounded treewidth k-planar Walczak [24] Excluded minor Example 2
Excluded apex minor
Bounded degree 6∆ F¨ uredi & Kahn [9]
Joret et al. [10]
Bounded genus Planar
Bounded treewidth
Streib & Trotter [21] Treewidth 6 2 + Pathwidth 6 2
dim(P ) 6 17 Bir´o et al. [1]
=
or
dim(P ) 6 1276 Joret et al. [11]
Outerplanar dim(P ) 6 4 Felsner et al. [8]
Forests dim(P ) 6 3 Trotter & Moore [23]
Figure 2. A hierarchy of sparse classes of graphs. A type of classes is drawn in grey whenever it includes a class C such that there are posets of bounded height and unbounded dimension with cover graphs in C.
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that is, minors that can be obtained by first contracting disjoint connected subgraphs of bounded radius, and then possibly removing some vertices or edges. In a nowhere dense class it is required that bounded-depth minors exclude at least some graph (which can depend on the depth). In a class with bounded expansion the requirement is stronger: For every r > 1, depth-r minors should be sparse, that is, their average degrees should be bounded by some function f (r). It is well-known that graphs with no H-minors have average degree bounded by a function of H. Thus the corresponding class of graphs has bounded expansion, since the average degree of their minors is uniformly bounded by a constant. The class of graphs with no topological H-minors also has bounded expansion, though in that case the bounding function might not be constant anymore, see [17]. These remarks show that Theorem 1 generalizes the aforementioned result for cover graphs excluding a fixed graph as a topological minor, and hence the previous body of work as well. In Figure 2 we give a summary of all the known results about posets with sparse cover graphs having their dimension bounded by a function of their height. We also mention the cases where dimension is bounded by an absolute constant, see the bottom part of the figure. Complementing our main [12] result, we describe in Section 2 a family of height-2 posets whose cover graphs form a nowhere dense class and having unbounded dimension. Thus Theorem 1 cannot be extended to nowhere dense classes. The graphs constructed moreover have locally bounded treewidth, showing that this property alone is not sufficient either. Finally, in order to complete our study of sparse cover graphs following the theory of Neˇsetˇril and Ossona de Mendez, we remark that another specialization of nowhere dense classes called almost wide classes also fail, in the sense that dimension is not bounded in terms of height for posets with cover graphs belonging to such a class. (We postpone the rather technical definition of almost wide classes until Section 2.)
1.4. Applications. Let us now turn our attention to some applications of Theorem 1 in the context of graph drawing, where natural classes with bounded expansion appear that do not fit in the previous setting of excluding a (topological) minor. Consider for instance posets whose diagrams can be drawn with ‘few’ edge crossings. If we bound the total number of crossings, then the cover graphs have bounded genus, and hence earlier results apply. On the other hand, if we only bound the number of crossings per edge in the drawing, say at most k such crossings, then we come to the well-studied class of kplanar graphs. While k-planar √ graphs are sparse—Pach and T´oth [20] proved that their average degree is at most 8 k— they do not exclude any graph as a topological minor (assuming k > 1), since every graph has a 1-planar subdivision: just start with any drawing of the graph in the plane and subdivide its edges around each crossing. Note however that this could require some edges to be subdivided many times. Indeed, one can observe that if an (< r)-subdivision of a graph G is k-planar then G is kr-planar. Combining this fact with Pach and T´oth’s result, Neˇsetˇril, Ossona de Mendez, and Wood [18] proved that the class of k-planar graphs has bounded expansion. Therefore, by Theorem 1, whenever a poset P has a k-planar cover graph, dim(P ) is bounded by a function of k and the height of P .
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Another example is given by graphs with bounded book thickness. A book embedding of a graph is a collection of half-planes (pages) all having the same line as their boundary (the spine) such that all vertices of the graph lie on the line, every edge is contained in one of the half-planes, and edges on a same page do not cross. The book thickness (also known as stack number ) of a graph is the smallest number of pages in a book embedding. Clearly, every graph on n vertices with book thickness k has at most 2kn edges. Still, every graph has a subdivision with book thickness at most 3 (see e.g. [4]). Neˇsetˇril et al. [18] observed that a result of Enomoto, Miyauchi, and Ota [6] easily implies that every class of graphs with bounded book thickness has bounded expansion. Therefore, by Theorem 1, whenever a poset P has a cover graph of book thickness at most k, its dimension is bounded by a function of k and the height of P . Yet another class of graphs with bounded expansion from graph drawing is that of graphs with bounded queue number. A queue layout of a graph is an ordering of its vertices (the spine) together with an edge coloring such that there are no two nested monochromatic edges, where two edges are nested if all four endpoints are distinct and the endpoints of one edge induce an interval on the spine containing the endpoints of the other edge. Then the queue number of a graph is the minimum number of colors in a queue layout. Every graph has a subdivision with queue number 2, as proved by Dujmovi´c and Wood [4], who also showed that graphs with bounded queue number form a class with bounded expansion. The challenging open problem in the area of queue layouts is to decide whether planar graphs have constant queue number (see e.g. [3]).
1.5. Proof ideas. We conclude this introduction with a brief overview of the two main tools used in the proof of Theorem 1. First, we use a characterization of classes with bounded expansion in terms of p-centered colorings. A p-centered coloring of a graph G is a vertex coloring of G such that, for every connected subgraph H of G, either some color is used exactly once in H, or at least p colors are used in H. Neˇsetˇril and Ossona de Mendez [15] proved that a class C has bounded expansion if and only if there is a function f such that for every integer p > 1 and every graph G ∈ C, there is a p-centered coloring of G using at most f (p) colors. Given a poset P of height h with cover graph in C, we work with a (2h − 1)-centered coloring of the cover graph with at most f (2h − 1) colors. Second, we make an exhaustive use of a decomposition for posets with large dimension called unrolling, and identify some ‘local’ part of the unrolling that still has large dimension. This simple idea, which is encapsulated in Lemma 7, is reminiscent of breadth-first search layerings for graphs and the observation that if the graph has large chromatic number then some layer still has large chromatic number. Given a poset and some (2h − 1)-centered coloring of its cover graph, we repeatedly apply the unrolling lemma as long as the dimension is large, each time winning an extra structure that is ‘floating around’ the remaining local poset with large dimension. At the very end, if enough iterations were performed (i.e. if the dimension was large enough), then we use this extra structure to find a connected subgraph of the cover graph where at most 2h − 2 colors are used and no color appears exactly once, contradicting the properties of the given (2h − 1)-centered coloring.
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We note that the idea of unrolling a poset was originally introduced by Streib and Trotter [21]. Very recently, Micek and Wiechert [14] used a similar iterative process to obtain an alternative proof of Walczak’s theorem for cover graphs in a class C excluding a fixed topological minor H: Essentially, they are able to find a subdivision of H in the cover graph when the process lasts too long. Unfortunately, there is no control on the depth of the H-minor constructed, and thus the approach of [14] cannot work directly for a class C with bounded expansion. Our proof of Theorem 1 avoids this difficulty by combining iterative unrolling with the powerful p-centered colorings. In fact, the proof is surprisingly short and simple, especially compared to previous works in the area. The paper is organized as follows. In Section 2 we give the necessary definitions for all sparse classes of graphs depicted in Figure 2, as well as the necessary background regarding posets. Along the way, we also describe constructions of posets showing that Theorem 1 cannot be extended to the classes in grey in Figure 2. We conclude that section with a key lemma about unrolling a poset with large dimension. In Section 3 we give a proof of Theorem 1.
2. Definitions and preliminaries 2.1. Posets. Let us start with basic notions about posets. All posets considered in this paper are finite. Elements of a poset P are called points. Points x, y ∈ P are said to be comparable in P if x 6 y or x > y in P . Otherwise x and y are incomparable in P . Given a poset P and a subset X of its points, the subposet of P induced by X is the poset on X where relations between the points are as in P . A set of points C ⊆ P is a chain in P if the points in C are pairwise comparable. The height of P is the maximal size of a chain in P . We write x < y in P if it holds that x 6 y and x 6= y. For distinct x, y ∈ P , if x < y in P and there is no z ∈ P with x < z < y in P then x < y is a cover relation of P . By cover(P ) we denote the cover graph of P , the (undirected) graph defined on the points of P where edges correspond to cover relations of P . A linear extension L of P is a poset on the points of P such that the points are pairwise comparable in L and x 6 y in L whenever x 6 y in P . Linear extensions L1 , . . . , Ld form a realizer of P if their intersection is equal to P , that is, x 6 y in P if and only if x 6 y in Li for each i ∈ {1, . . . , d}. The dimension of P , denoted by dim(P ), is the least number d such that there is a realizer of cardinality d of P .
2.2. Sparse graph classes. Next, we introduce the necessary definitions regarding graphs and give proper definitions for the graph classes mentioned in the introduction. All graphs in this paper are finite, simple, and undirected. Given a graph G we denote by V (G) and E(G) the vertex set and edge set of G, respectively. For a subset X ⊆ V (G) we denote by G[X] the subgraph of G induced by vertices in X. The distance between two vertices in G is the length of a shortest path between them. (Thus adjacent vertices are at distance 1; also, distance between two vertices in distinct components of G is set to +∞.) The set of all vertices at distance at most r from vertex v in G is denoted by NGr (v), and the subscript is omitted if G is clear from the context. The radius of a
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connected graph G is the least integer r > 0 for which there is a vertex v ∈ V (G) such that NGr (v) = V (G). The treewidth of a graph G, denoted by tw(G), is the least integer t > 0 for which there is a tree T and a family {T (v) | v ∈ V (G)} of non-empty subtrees of T such that |{v ∈ V (G) | x ∈ T (v)}| 6 t + 1 for each node x of T , and V (T (u)) ∩ V (T (v)) 6= ∅ for each edge uv ∈ E(G). A class of graphs C has locally bounded treewidth if there exists a function f : R → R such that tw(G[N r (v)]) 6 f (r) for every integer r > 0, graph G ∈ C and vertex v ∈ V (G). Given a partition X of the vertices of a graph G into non-empty parts inducing connected subgraphs, we denote by G/X the graph with vertex set X and edge set defined as follows: For two distinct distinct parts X, Y ∈ X , we have XY ∈ E(G/X ) if and only if there exist x ∈ X and y ∈ Y such that xy ∈ E(G). A graph H is a minor of G if H is isomorphic to a subgraph of G/X for some such partition X of V (G). A specialization of this notion is that of topological minors: H is a topological minor of a graph G if G contains a subgraph isomorphic to a subdivision of H. A class of graphs C is minor closed (topologically closed ) if every minor (topological minor, respectively) of a graph in C is also in C. We pursue with the definitions of classes with bounded expansion and nowhere dense classes. A graph H is a depth-r minor (also known as an r-shallow minor ) of a graph G if H is isomorphic to a subgraph of G/X for some partition X of V (G) into nonempty parts inducing subgraphs of radius at most r. The greatest reduced average density n (grad ) of rank r of a graph G,o denoted by ∇r (G), is defined as ∇r (G) = |E(H)| max |V | H is a depth-r minor of G . A class of graphs C has bounded expansion (H)| if there exists a function f : R → R such that ∇r (G) 6 f (r) for every integer r > 0 and graph G ∈ C. A class of graphs C is nowhere dense if for each integer r > 0 there exists a graph which is not a depth-r minor of any graph G ∈ C. It is easy to see that classes with locally bounded treewidth and classes with bounded expansion are nowhere dense. Note however that these two notions are incomparable. Two classical examples of classes with locally bounded treewidth are graphs with bounded maximum degree, and graphs excluding some apex graph A as a minor. (Recall that A is apex if A can be made planar by removing at most one vertex.) The latter is in fact a characterization of minor-closed classes with locally bounded treewidth: A minor-closed class C has locally bounded treewidth if and only if C excludes some apex graph, a fact originally proved by Eppstein [7]. Let us now define almost wide classes. For d > 1, a set of vertices X in a graph G is dindependent if every two distinct vertices in X are at distance strictly greater than d in G. A class of graphs C is almost wide if there exists an integer s > 0 such that for every integer d > 1 there is a function f : R → R such that for every integer m > 1, every graph G ∈ C of order at least f (m) contains a subset S of size at most s so that G − S has a d-independent set of size m. Neˇsetˇril and Ossona de Mendez [16, Theorem 3.23] proved that a class of graphs excluding a fixed graph as a topological minor is almost
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wide. Moreover, they proved [16, Theorem 3.13] that every hereditary class of graphs that is almost wide is also nowhere dense. (Recall that a class is hereditary if it is closed under taking induced subgraphs.) 2.3. Posets with cover graphs in a nowhere dense class. As mentioned in the introduction, the statement of Theorem 1 cannot be pushed further towards nowhere dense classes. In fact, it already fails for classes with locally bounded treewidth and hereditary classes that are almost wide, as we now show. Our construction is based on the class of graphs G with ∆(G) 6 girth(G), where ∆(G) and girth(G) denote the maximum degree and girth of G, respectively. This is a useful example of a hereditary class with locally bounded treewidth that is also almost wide but that has not bounded expansion. (Indeed, this class is invoked several times in the textbook [17].) We will use in particular that the chromatic number of these graphs is unbounded. This is a well-known fact that can be shown in multiple ways; we can note for instance that the chromatic number of the n-vertex d-regular non-bipartite Ramanujan graphs with girth √ Ωd (log n) built by Lubotzky, Phillips, and Sarnak [13] have chromatic number Ω( d) (see [13]). Proposition 2. There exists a hereditary almost wide class of graphs C with locally bounded treewidth such that posets of height 2 with cover graphs in C have unbounded dimension. Proof. For a graph G, the adjacency poset PG of G is the poset with point set {av | v ∈ V (G)} ∪ {bv | v ∈ V (G)} such that, for every two distinct vertices u, v ∈ V (G), we have au 6 bv in PG if and only if uv ∈ E(G). It is well known that dim(PG ) > χ(G). This can be seen as follows. Fix a realizer L1 , . . . , Ld of PG . For every vertex v ∈ V (G), fix a number φ(v) = i such that bv < av in Li . We claim that φ is a proper coloring of G. Consider any two adjacent vertices u and v in G and, say, φ(v) = i. Then au 6 bv < av 6 bu in Li , which witnesses that φ(u) 6= i. Therefore, dim(PG ) > χ(G). Now let C denote the class of graphs G satisfying ∆(G) 6 girth(G). As mentioned earlier, this class is hereditary, almost wide, and has locally bounded treewidth. This is not difficult to check (or see [17] for a proof). The key observation about the class C in this context is that if G ∈ C, then cover(PG ) ∈ C. This can be seen as follows: First, clearly ∆(G) = ∆(cover(PG )), so it is enough to show girth(G) 6 girth(cover(PG )). To show the latter, we remark that if C is a cycle of cover(PG ), then C naturally corresponds to a closed walk W in G of the same length. Moreover, every three consecutive vertices in that walk W are pairwise distinct, as follows from the adjacency poset construction. Hence, W contains a cycle, which is of length at most that of C. Therefore, girth(G) 6 girth(cover(PG )), as claimed. To summarize, graphs in C have unbounded chromatic number, implying that adjacency posets of these graphs have unbounded dimension. Yet, the cover graphs of these adjacency posets all belong to C, a hereditary almost wide class with locally bounded treewidth. This concludes the proof.
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2.4. Posets with cover graphs in a class with bounded expansion. There are a number of equivalent conditions for classes of graphs to have bounded expansion (see [19] for instance). As described in the introduction we make use of a characterization in terms of p-centered colorings. Recall that for a given class C with bounded expansion there is a function f such that for every integer p > 1 and every graph G ∈ C there is a p-centered coloring of G using at most f (p) colors. Therefore, the following theorem implies Theorem 1. Theorem 3. If P is a poset of height h such that its cover graph has a (2h − 1)-centered coloring using c colors, then 2 2 !3h2 (cˆ+1 h ) cˆ + 1 , dim(P ) 6 2 · 2h2 h where cˆ = max{c, 2h − 2}. The proof of Theorem 3 is given in Section 3. As a side remark, if we relax the requirement in Theorem 3 by replacing the (2h − 1)-centered coloring with a (2h − 2)-centered coloring then the statement becomes false: Proposition 4. For every h > 2 and c > 2h − 2, posets of height h whose cover graphs admit a (2h − 2)-centered coloring with at most c colors have unbounded dimension. Proof. Clearly, it is enough to show the claim for c = 2h − 2. Fix h > 2. Let Pn (n > 3) be the poset obtained from the incidence poset IKn of Kn by subdividing every comparability v < e in IKn with h − 2 distinct points; that is, points x1 , . . . , xh−2 are added such that v < x1 < · · · < xh−2 < e in Pn , and this chain displays all of their comparabilities in Pn . Note that Pn has height h. Recall that dim(IKn ) > log log(n), as discussed in the introduction. Since Pn contains IKn as an induced subposet, the dimension of Pn grows with n. Note also that the cover graph of Pn is the complete graph Kn where each edge has been subdivided exactly 2h − 3 times. Let Jn denote this graph. It remains to show that Jn has a (2h−2)-centered coloring. For this we define a coloring φ of Jn . Let V be the set of vertices of Jn with degree bigger than 2. We set φ(v) = 0 for each vertex v ∈ V , and for every two distinct vertices u, v ∈ V we let φ use each color from {1, . . . , 2h − 3} exactly once for the 2h − 3 internal vertices of the u–v path in Jn that comes from the subdivision of the edge uv. Clearly, φ uses 2h − 2 colors in total. To show that φ is (2h − 2)-centered, consider a connected subgraph H of Jn . If H avoids all vertices in V then H is a subpath of some u–v path considered earlier. In this case, all vertices in H have different colors by construction. If |V (H) ∩ V | = 1, then exactly one vertex of H is colored with 0. Finally, if |V (H) ∩ V | > 2, then H contains some u–v path as above as a subgraph, thus all 2h − 2 colors appear on H. We conclude that φ is a (2h − 2)-centered coloring. 2.5. Tools for dimension. We now give some preliminary lemmas regarding posets and their dimension. We let Inc(P ) = {(x, y) ∈ P × P | x is incomparable with y in P }
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denote the set of ordered pairs of incomparable points in P . We say that a point x ∈ P is minimal (maximal ) if there is no z ∈ P with z < x in P (x < z in P ). We denote by Min(P ) the set of minimal points in P and by Max(P ) the set of maximal points in P . The downset of a set S ⊆ P of points is defined as D(S) = {x ∈ P | ∃s ∈ S such that x 6 s in P }, and similarly we define the upset of S to be U(S) = {x ∈ P | ∃s ∈ S such that s 6 x in P }. A set I ⊆ Inc(P ) of incomparable pairs is reversible if there is a linear extension L of P that reverses each pair in I, that is y < x in L for every (x, y) ∈ I. Rephrasing the definition of dimension, dim(P ) is the least positive integer d for which there exists a partition of Inc(P ) into d reversible sets if Inc(P ) is not empty, and defined as 1 otherwise. For sets A ⊆ Min(P ) and B ⊆ Max(P ) let IncP (A, B) = {(a, b) ∈ Inc(P ) | a ∈ A and b ∈ B}. If IncP (A, B) is not empty then we define dimP (A, B) to be the least d such that IncP (A, B) can be partitioned into d reversible sets of P . Otherwise, we set dimP (A, B) = 0. We write dim(A, B) instead of dimP (A, B) when the poset is clear from the context. We use the shorthand [k] := {1, . . . , k}. Observation 5. Let P be a poset and A ⊆ Min(P ), B ⊆ Max(P ). (i) If Q is an induced subposet of P with A ⊆ Min(Q) and B ⊆ Max(Q), then dimQ (A, B) = dimP (A, B). (ii) If B1 , . . . , Bk is a partition of B, then X dimP (A, B) 6 dimP (A, Bi ), i∈[k]
and consequently there is ` ∈ [k] such that dimP (A, B` ) > dimP (A, B)/k. Proof. Item (i) follows from the observation that the restriction of any linear extension of P to the points of Q gives a linear extension of Q, and converserly, any linear extension of Q can be extended to a linear extension of P . (Note that it is crucial that Q is an induced subposet of P .) To show Item (ii), let Li (i ∈ [k]) denote the set of linear extensions of P witnessing dimP (A, Bi ).P Then L1 ∪ · · · ∪ Lk reverses all incomparable pairs in IncP (A, B) and has size at most i∈[k] dimP (A, Bi ). In our main proof we will focus on reversing incomparable pairs consisting of a minimal and a maximal point. The following observation is standard, see [11] for a proof. Observation 6. For every poset P there is a poset Q such that (i) height(Q) = height(P ), (ii) cover(Q) can be obtained from cover(P ) by attaching vertices of degree 1, and (iii) dim(P ) 6 dim(Min(Q), Max(Q)).
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B3
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A2
A1 a0
Figure 3. Unrolling of P starting from a0 . Suppose P is a connected poset, that is, that the cover graph of P is connected. Let A = Min(P ) and let B = Max(P ). Choose arbitrarily a0 ∈ A and set A0 = {a0 }. For i = 1, 2, . . . let n o [ Bi = b ∈ B − Bj | there is a ∈ Ai−1 with a 6 b in P , 16j
n o [ Ai = a ∈ A − Aj | there is b ∈ Bi with a 6 b in P . 16j
Let m be the least index such that Am is empty. Since P is connected, the sets A0 , . . . , Am−1 partition A, and the sets B1 , . . . , Bm partition B. We say that the sequence A0 , B1 , . . . , Am−1 , Bm is obtained by unrolling P from a0 . See an illustration of this decomposition in Figure 3. Also note a useful property of this construction: for every a ∈ Ai and b ∈ B with a 6 b in P we have b ∈ Bi ∪ Bi+1 , (?) for every b ∈ Bi and a ∈ A with a 6 b in P we have a ∈ Ai−1 ∪ Ai .
The following lemma intuitively says that each unrolling contains a heavy part with respect to dimension. We refer the reader to [14] for a proof. Lemma 7. Let P be a connected poset, let A = Min(P ), B = Max(P ), and suppose that dim(A, B) > 2. Consider a sequence A0 , B1 , . . . , Am−1 , Bm obtained by unrolling P . Then there is an index ` ∈ [m − 1] such that dim(A` , B` ) > dim(A, B)/2
or
dim(A` , B`+1 ) > dim(A, B)/2.
3. Proof of Theorem 3 In this section we give a proof of Theorem 3. Let thus P be a poset of height h, and let φ be a (2h − 1)-centered coloring of the cover graph of P using colors from the set {1, . . . , c}. If h = 1 then P is an antichain and dim(P ) 6 2. So we may assume h > 2. We may also suppose that c > 2h − 2, as otherwise we add artificial colors to the color set without affecting φ being (2h − 1)-centered. Thus, the inequality we need to prove is: 2 2 !3h2 (c+1 h ) c+1 dim(P ) 6 2 · 2h2 . (1) h First we apply Observation 6 to obtain a poset P 0 of height h with dim(P ) 6 dim(Min(P 0 ), Max(P 0 )) and such that cover(P 0 ) is obtained from cover(P ) by attaching
SPARSITY AND DIMENSION
13
some degree-1 vertices. We extend φ to the graph cover(P 0 ) by assigning to all new vertices a new color, namely c + 1. We claim that φ is a (2h − 1)-centered coloring of cover(P 0 ). Indeed, consider any connected subgraph H of cover(P 0 ). Since all new vertices are of degree 1, either H ∩ cover(P ) is non empty and connected, or H consists of just a single new vertex of color c + 1. Note that in the first case, either φ uses at least 2h − 1 colors on H ∩ cover(P ) (and in particular on H), or there is a color used exactly once on H ∩ cover(P ), and this color being distinct from c + 1 is also unique on H. Hence, φ is also a (2h − 1)-centered coloring of cover(P 0 ). We will focus on P 0 and show that the right-hand side of (1) is an upper bound on dim(Min(P 0 ), Max(P 0 )), which is enough since dim(P ) 6 dim(Min(P 0 ), Max(P 0 )). With a slight abuse of notation, and since we will not need to consider the old poset P anymore, for convenience we write P instead of P 0 from now on. Let us proceed with a short overview of the proof. We set up an iterative process that maintains two sets A ⊆ Min(P ) and B ⊆ Max(P ). Initially we set A := Min(P ) and B := Max(P ), and thus dim(A, B) = dim(Min(P ), Max(P )) at the beginning. We proceed with a new iteration as long as the current sets A and B satisfy dim(A, B) > 3. An iteration consists in finding non-empty subsets A0 ⊆ A and B 0 ⊆ B by unrolling the current poset (this will be explained shortly) that have some useful structural properties, and satisfying −2 dim(A, B) c+1 0 0 dim(A , B ) > . (2) · h 2 h (Note that, since dim(A0 , B 0 ) is an integer, we always have dim(A0 , B 0 ) > 1.) Then, at the end of the iteration, we update the sets A and B by setting A := A0 and B := B 0 . Every iteration has a corresponding type, a 3-tuple (α, X, Y ) where α ∈ {1, 2, 3} encodes the unrolling type (to be defined later), and X and Y are non-empty subsets of the color 2 set {1, . . . , c + 1}, each of size at most h. Thus, there are at most 3h2 c+1 different h iteration types (since c > 2h − 2). A key property of our approach will be that no two iterations can have the same type; this will be shown at the end of the proof. Note that 2 this implies that after at most 3h2 c+1 iterations we must have dim(A, B) 6 2, which h gives the desired bound on dim(Min(P ), Max(P )). We continue with a precise description of an iteration and the definition of its type. Suppose we are given A ⊆ Min(P ) and B ⊆ Max(P ) with dim(A, B) > 3. First, we clean the picture by ruling out some trivialities. It is well-known that the dimension of a poset is witnessed by the dimension of a subposet induced by a single component of the cover graph. (To be precise, this holds except when the poset is a union of chains.) In our setting, we are interested in bounding dim(A, B). Using that dim(A, B) > 3, we can find sets A00 ⊆ A and B 00 ⊆ B such that • dim(A00 , B 00 ) = dim(A, B); • U(A00 ) ∩ D(B 00 ) induces a connected subgraph of cover(P ), and • A00 ⊆ D(B 00 ) and B 00 ⊆ U(A00 ).
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(We leave the easy proof to the reader.) Define Q to be the subposet of P induced by U(A00 ) ∩ D(B 00 ). Thus, Q is connected, A00 = Min(Q), and B 00 = Max(Q). By Observation 5(i), we have dimQ (A00 , B 00 ) = dimP (A00 , B 00 ). Choose a point a0 ∈ A00 arbitrarily, and let A0 , B1 , A1 , . . . , Am−1 , Bm be the sequence obtained by unrolling Q from a0 . By Lemma 7 there is an index ` ∈ [m − 1] such that dimQ (A` , B` ) > dimQ (A, B)/2 or dimQ (A` , B`+1 ) > dimQ (A, B)/2. Since Q is an induced subposet of P , this implies dimP (A` , B` ) > dimP (A, B)/2
or
dimP (A` , B`+1 ) > dimP (A, B)/2.
These two possibilities are responsible for the first coordinate of an iteration type, the unrolling type: If dimP (A` , B` ) > dimP (A, B)/2 then the unrolling type is 1 or 2, depending on whether ` = 1 or ` > 2. Otherwise, we have dimP (A` , B`+1 ) > dimP (A, B)/2, and the unrolling type is 3. We now consider each of the three unrolling types separately. Unrolling type 1. Suppose that the unrolling type is 1, thus dimP (A1 , B1 ) > dimP (A, B)/2. Observe that a0 < b in Q for all points b ∈ B1 in this case. Suppose that the current iteration is the i-th iteration in our process. For every point b ∈ B1 , we fix a chain C i (b) = {y1 , . . . , yr } of Q with y1 = a0 and yr = b, and such that yk < yk+1 is a cover relation in P for each k ∈ [r−1]. The chain has length at most h as this is the height of P . Thus, the set of colors φ(C i (b)) is a non-empty subset of [c + 1] of size at most h, and hence there are at most h c+1 possible values h i of φ(C (b)) (as c > 2h − 2). These values induce a partition of B1 into at most h c+1 h sets. Using Observation 5(ii), it follows that for some Y ⊆ [c + 1] of size at most h we −1 have that B 0 := {b ∈ B1 | φ(C i (b)) = Y } satisfies dimP (A1 , B 0 ) > dimP2(A,B) · h c+1 . h Note that points of A1 that are below all the points from B 0 in P do not contribute to dimP (A1 , B 0 ) as they are not involved in any pair in IncP (A1 , B 0 ). We set A0 := A1 − {a ∈ A1 | a 6 b in P for all b ∈ B 0 }, and thus have −1 dimP (A, B) c+1 0 0 0 dimP (A , B ) = dimP (A1 , B ) > · h . 2 h Clearly, A0 and B 0 are non empty sets since otherwise we would have dim(A0 , B 0 ) = 0, while the right-hand side of the inequality is strictly greater than 0. We define the iteration type in this case to be (1, ∅, Y ), and note that the desired lower bound on dimP (A0 , B 0 ) (c.f. (2)) is satisfied. Let us summarize some important properties that we have in this case: (1.1) for every b ∈ B 0 , the points of C i (b) form a chain with b = max(C i (b));
SPARSITY AND DIMENSION
(1.2) (1.3) (1.4) (1.5)
15
for every b ∈ B 0 , the chain C i (b) induces a connected subgraph of cover(P ); for every b ∈ B 0 , we have φ(C i (b)) = Y ; there exists a ∈ A − A0 such that min(C i (b)) = a for every b ∈ B 0 , and for every a ∈ A0 , there exists b ∈ B 0 such that a and b are incomparable in P .
Unrolling type 2. Suppose next that the unrolling type is 2, so dimP (A` , B` ) > dimP (A, B)/2 for some ` > 2. In particular, the sets A`−1 and B`−1 exist. Recall that, by the definition of unrolling Q, every point in B` is above some point in A`−1 in Q, and also every point in A`−1 is below some point in B`−1 in Q. For every point b ∈ B` we define the following sets: First, we fix a chain b = y1 > · · · > yj = a of Q such that a ∈ A`−1 and yk > yk+1 is a cover relation in P for every k ∈ [j − 1]. Let r ∈ [j − 1] be minimal such that yr ∈ D(B`−1 ), which is well defined since a = yj ∈ D(B`−1 ). (The two downsets are with respect to poset Q.) Next, we fix a chain yr = x1 < x2 < · · · < xs in Q which is such that xs ∈ B`−1 and xk < xk+1 is a cover relation in P for every k ∈ [s − 1]. Now we wish to associate two chains to the point b. For this suppose that we are currently in the i-th iteration. Then let Li (b) = {x1 , . . . , xs } and Ri (b) = {y1 , . . . , yr−1 }. Since Li (b) and Ri (b) form chains of 2 size at most h and h − 1 in P , respectively, there at most h c+1 possibilities for the h i i pair (φ(L (b)), φ(R (b))). Considering the value of the pair (φ(Li (b)), φ(Ri (b))) for each point b ∈ B` , we obtain a 2 partition of B` into at most h c+1 sets. As a consequence, there are sets X, Y ⊆ [c+1] h of size at most h and h − 1, respectively, such that B 0 := {b ∈ B` | φ(Li (b)) = X and φ(Ri (b)) = Y } satisfies −2 dimP (A, B) c+1 0 dimP (A` , B ) > · h . 2 h We set A0 := A` and define the iteration type to be (2, X, Y ). Note again that A0 and B 0 are non empty, and that inequality (2) is satisfied. It is straightforward to verify the following properties: For every point b ∈ B 0 , points in Li (b) and Ri (b) form chains in P such that (2.1) (2.2) (2.3) (2.4) (2.5) (2.6)
max(Li (b)) ∈ B − B 0 and max(Ri (b)) = b; Li (b) ∪ Ri (b) induces a connected subgraph of cover(P ); the upset of Ri (b) in P avoids points of Li (b); the upset of A0 in P avoids points of Li (b) (by the property (?) of unrolling); the upset of A in P contains Li (b), and φ(Li (b)) = X and φ(Ri (b)) = Y , and |X| 6 h and |Y | 6 h − 1.
Unrolling type 3. Assume now that the unrolling type is 3. Then dim(A` , B`+1 ) > dim(A, B)/2 for some ` > 2. Note that ` > 1 since dim(A0 , B1 ) = 0. In particular, the sets B` and A`−1 exist. We will proceed analogously to the previous case.
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For every a ∈ A` we define the following sets: First, we fix a chain a = y1 < y2 < · · · < yr = b of Q such that b ∈ B` and yk < yk+1 is a cover relation in P for every k ∈ [r − 1]. Let j ∈ [r − 1] be minimal such that yj ∈ U(A`−1 ). Next, we fix a chain x1 < x2 < · · · < xs = yj of Q with x1 ∈ A`−1 and with xk < xk+1 being a cover relation in P for every k ∈ [s − 1]. Suppose we are in the i-th iteration. Then we set Li (a) = {x1 , . . . , xs } and Ri (a) = {y1 , . . . , yj−1 }. 2 As before, for a point a ∈ A` , there are at most h c+1 possible evaluations of the h 2 i i pair (φ(L (a)), φ(R (a))). This induces a partition of A` into at most h c+1 sets. We h deduce in turn that there are sets X, Y ⊆ [c+1] of size at most h and h−1, respectively, such that A0 := {a ∈ A` | φ(Li (a)) = X and φ(Ri (a)) = Y } satisfies −2 dim(A, B) c+1 0 dim(A , B`+1 ) > · h . 2 h We set B 0 := B`+1 and define the iteration type to be (3, X, Y ). Clearly, A0 and B 0 are non empty, and inequality (2) is satisfied. This concludes our study of the three unrolling types. Iteration types do not reoccur. Now we will prove that no two iterations have the same type. We begin with iterations that have unrolling types 1. Claim. For each Y ⊆ [c + 1], there is at most one iteration with type (1, ∅, Y ). Proof. Suppose for a contradiction that two different iterations have the same type (1, ∅, Y ), say the i-th and j-th iterations, with i < j. Let Ai , B i and Aj , B j denote the value of the subsets A, B at the end iteration i and j, respectively. Then Aj ⊆ Ai and B j ⊆ B i . Let ai and aj be points witnessing property (1.4) for iteration i and j respectively (notice that in each case it suffices to take the point from which the current poset Q was unrolled). By construction dim(Aj , B j ) > 1, and hence B j is not empty. Choose arbitrarily a point b ∈ B j ⊆ B i and let b0 ∈ B i be such that (aj , b0 ) ∈ Inc(Ai , B i ) (such a b0 exists by (1.5)). Now consider the three chains C i (b0 ), C i (b), and C j (b); see Figure 4 (left) for an illustration. Since C i (b0 ) and C i (b) both contain ai by (1.4), and C i (b) and C j (b) both contain b, it follows by (1.2) that the union U of the three chains induce a connected subgraph of cover(P ). As iterations i and j both have type (1, ∅, Y ), by (1.3) we have φ(C i (b0 )) = φ(C i (b)) = φ(C j (b0 )) = Y , and hence at most h colors are used on U . Furthermore, each color in Y is used at least twice on U . This is because the chains C i (b0 ) and C j (b) must be disjoint, since otherwise aj and b0 would be comparable in P . However, since h < 2h − 1, this contradicts the fact that φ is a (2h − 1)-centered coloring of cover(P ). Claim. For each X, Y ⊆ [c + 1] there is at most one iteration of type (2, X, Y ). Proof. Suppose that there are two different iterations of type (2, X, Y ). Say this is the case for the i-th and j-th iterations, with i < j. Let Ai , B i and Aj , B j denote the
SPARSITY AND DIMENSION b0
Bi
ai
b
Bi
Bj
aj
Ai
Aj
b0
Li (b0 ) Ri (b0 )
17 Bj
Ai
b
Aj
Figure 4. Left: union of the three chains C i (b0 ), C i (b), and C j (b). Right: union of the four chains Li (b0 ), Ri (b0 ), Lj (b), and Rj (b). Dotted lines indicate incomparabilities. value of the subsets A, B at the end iteration i and j, respectively. Then Aj ⊆ Ai and B j ⊆ B i . By construction dim(Aj , B j ) > 1, and hence B j is not empty. Fix a point b ∈ B j and consider the two chains Lj (b) and Rj (b). Let b0 = max(Lj (b)), which is contained in set B i by (2.1). For the rest of the argument we focus on the four chains Li (b0 ), Ri (b0 ), Lj (b), Rj (b), and their union U = Li (b0 ) ∪ Ri (b0 ) ∪ Lj (b) ∪ Rj (b); see Figure 4 (right) for an illustration. Since iterations i and j share type (2, X, Y ), by (2.6) we have X = φ(Li (b0 )) = φ(Lj (b)), Y = φ(Ri (b0 )) = φ(Rj (b)), and hence φ(U ) = X ∪ Y . Observe that U induces a connected subgraph of cover(P ). Indeed, unions Li (b0 ) ∪ Ri (b0 ) and Lj (b) ∪ Rj (b) are connected in cover(P ) by (2.2), and both unions share point b0 . Furthermore, since b0 is contained in Lj (b) and Ri (b0 ), we conclude that color sets X and Y are not disjoint. Together with (2.6) this implies |X ∪ Y | 6 2h − 2. Now note that Li (b0 ) and Lj (b) cannot intersect since the upset of Ai in P is disjoint from Li (b0 ) (by (2.4) for iteration i), but it contains Lj (b) (by (2.5) for iteration j). Hence no color from X is unique on U . Chains Ri (b0 ) and Rj (b) cannot intersect as well since otherwise b0 = max(Ri (b0 )) would be in the upset of Rj (b). However, this contradicts (2.3) since b0 is contained in Lj (b). It follows that no color from Y is unique on U either. This contradicts the fact that φ is (2h − 1)-centered. It remains to consider the case of two different iterations sharing a type that is of the form (3, X, Y ). However, the argument goes along similar lines as for the type (2, X, Y ). Therefore, we omit a detailed treatment here. This conludes the proof of Theorem 3.
Acknowledgements The first author thanks David R. Wood for asking whether Theorem 1 holds in the case of cover graphs with bounded queue number, which prompted this line of research.
References [1] Csaba Bir´ o, Mitchel T. Keller, and Stephen J. Young. Posets with cover graph of pathwidth two have bounded dimension. arXiv:1308.4877.
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[2] Parinya Chalermsook, Bundit Laekhanukit, and Danupon Nanongkai. Graph products revisited: tight approximation hardness of induced matching, poset dimension and more. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1557–1576. SIAM, Philadelphia, PA, 2012. [3] Vida Dujmovi´c. Graph layouts via layered separators. J. Combin. Theory Ser. B, 110:79–89, 2015. arXiv:1302.0304. [4] Vida Dujmovi´c and David R. Wood. Stacks, queues and tracks: layouts of graph subdivisions. Discrete Math. Theor. Comput. Sci., 7(1):155–201, 2005. [5] Ben Dushnik and Edwin W. Miller. Partially ordered sets. Amer. J. Math., 63:600–610, 1941. [6] Hikoe Enomoto, Miki Shimabara Miyauchi, and Katsuhiro Ota. Lower bounds for the number of edge-crossings over the spine in a topological book embedding of a graph. Discrete Appl. Math., 92(2-3):149–155, 1999. [7] David Eppstein. Diameter and treewidth in minor-closed graph families. Algorithmica, 27(3-4):275– 291, 2000. arXiv:math/9907126. [8] Stefan Felsner, William T. Trotter, and Veit Wiechert. The Dimension of Posets with Planar Cover Graphs. Graphs Combin., 31(4):927–939, 2015. [9] Zoltan F¨ uredi and Jeff Kahn. On the dimensions of ordered sets of bounded degree. Order, 3(1):15– 20, 1986. [10] Gwena¨el Joret, Piotr Micek, Kevin G. Milans, William T. Trotter, Bartosz Walczak, and Ruidong Wang. Tree-width and dimension. Combinatorica, to appear. arXiv:1301.5271. [11] Gwena¨el Joret, Piotr Micek, William T. Trotter, Ruidong Wang, and Veit Wiechert. On the dimension of posets with cover graphs of treewidth 2. Submitted, arXiv:1406.3397. [12] David Kelly. On the dimension of partially ordered sets. Discrete Math., 35:135–156, 1981. [13] Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. Ramanujan graphs. Combinatorica, 8(3):261–277, 1988. [14] Piotr Micek and Veit Wiechert. Topological minors of cover graphs and dimension. submitted, arXiv:1504.07388. [15] Jaroslav Neˇsetˇril and Patrice Ossona de Mendez. Grad and classes with bounded expansion. I. Decompositions. European J. Combin., 29(3):760–776, 2008. arXiv:math/0508323. [16] Jaroslav Neˇsetˇril and Patrice Ossona de Mendez. First order properties on nowhere dense structures. J. Symbolic Logic, 75(3):868–887, 2010. [17] Jaroslav Neˇsetˇril and Patrice Ossona de Mendez. Sparsity, volume 28 of Algorithms and Combinatorics. Springer, Heidelberg, 2012. Graphs, structures, and algorithms. [18] Jaroslav Neˇsetˇril, Patrice Ossona de Mendez, and David R. Wood. Characterisations and examples of graph classes with bounded expansion. European J. Combin., 33(3):350–373, 2012. arXiv:0902.3265. [19] Jaroslav Neˇsetˇril, Patrice Ossona de Mendez, and David R. Wood. Characterisations and examples of graph classes with bounded expansion. European J. Combin., 33(3):350–373, 2012. [20] J´ anos Pach and G´eza T´ oth. Graphs drawn with few crossings per edge. Combinatorica, 17(3):427– 439, 1997. [21] Noah Streib and William T. Trotter. Dimension and height for posets with planar cover graphs. European J. Combin., 35:474–489, 2014. [22] William T. Trotter. Partially ordered sets. In Handbook of combinatorics, Vol. 1, 2, pages 433–480. Elsevier Sci. B. V., Amsterdam, 1995. [23] William T. Trotter, Jr. and John I. Moore, Jr. The dimension of planar posets. J. Combinatorial Theory Ser. B, 22(1):54–67, 1977. [24] Bartosz Walczak. Minors and dimension. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015), pages 1698–1707, 2015. arXiv:1407.4066. [25] Mihalis Yannakakis. The complexity of the partial order dimension problem. SIAM J. Algebraic Discrete Methods, 3(3):351–358, 1982.
arXiv:1507.01120v1 [math.CO] 4 Jul 2015
¨ JORET, PIOTR MICEK, AND VEIT WIECHERT GWENAEL
Abstract. We prove that posets of bounded height whose cover graphs belong to a fixed class with bounded expansion have bounded dimension. Bounded expansion, introduced by Neˇsetˇril and Ossona de Mendez as a model for sparsity in graphs, is a property that is naturally satisfied by a wide range of graph classes, from graph structure theory (graphs excluding a minor or a topological minor) to graph drawing (e.g. graphs with constant book thickness). Therefore, our theorem generalizes a number of results including the most recent one for posets of bounded height with cover graphs excluding a fixed graph as a topological minor (Walczak, SODA 2015). We also show that the result is in a sense best possible, as it does not extend to nowhere dense classes; in fact, it already fails for cover graphs with locally bounded treewidth.
´ Libre de Bruxelles, Brussels, (G. Joret) Computer Science Department, Universite Belgium (P. Micek) Theoretical Computer Science Department, Faculty of Mathematics and ´ w, Poland and Institut fu ¨ r MatheComputer Science, Jagiellonian University, Krako ¨ t Berlin, Berlin, Germany matik, Technische Universita ¨ r Mathematik, Technische Universita ¨ t Berlin, Berlin, (V. Wiechert) Institut fu Germany E-mail addresses: [email protected], [email protected], [email protected]. Date: July 7, 2015. 2010 Mathematics Subject Classification. 06A07, 05C35. Key words and phrases. Bounded expansion, poset, dimension, cover graph, graph minor. V. Wiechert is supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK 1408). 1
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G. JORET, P. MICEK, AND V. WIECHERT
1. Introduction 1.1. Poset dimension and cover graphs. Partially ordered sets, posets for short, are studied extensively in combinatorics, set theory and theoretical computer science. One of the most important measures of complexity of a poset is its dimension. The dimension dim(P ) of a poset P is the least integer d such that points of P can be embedded into Rd in such a way that x < y in P if and only if the point of x is below the point of y with respect to the product order on Rd . Equivalently, the dimension of P is the least d such that there are d linear extensions of P whose intersection is P . Not surprisingly, dimension is hard to compute: Already deciding whether dim(P ) 6 3 is an NP-hard problem [25], and for every ε > 0, there is no n1−ε -approximation algorithm for dimension unless ZPP=NP, where n denotes the size of the input poset [2]. Research in this area typically focus on finding witnesses for large dimension and sufficient conditions for small dimension; see e.g. [22] for a survey. Posets are visualized by their diagrams: Points are placed in the plane and whenever a < b in the poset, and there is no point c with a < c < b, there is a curve from a to b going upwards (that is y-monotone). The diagram represents those relations which are essential in the sense that they are not implied by transitivity, known as cover relations. The undirected graph implicitly defined by such a diagram is the cover graph of the poset. That graph can be thought of as encoding the ‘topology’ of the poset. There is a common belief that posets having a nice or well-structured drawing should have small dimension. But first let us notea negative observation by Kelly [12], from 1981, there is a family of posets whose diagrams can be drawn without edge crossings— undoubtedly qualifying as a ‘nice’ drawing—and with arbitrarily large dimension, see Figure 1. A key observation about Kelly’s construction is that these posets also have large height. This leads us to our main theorem, which generalizes several previous works in this area. Theorem 1. For every class of graphs C with bounded expansion, and for every integer h > 1, posets of height h whose cover graphs are in C have bounded dimension. 1.2. Background. Theorem 1 takes its roots in the following result of Streib and Trotter [21]: Posets with planar cover graphs have dimension bounded in terms of their height. Note that this justifies the fact that height and dimension grow together in Kelly’s construction. Joret, Micek, Milans, Trotter, Walczak, and Wang [10] subsequently proved that the same result holds in the case of cover graphs of bounded treewidth, of bounded genus, and more generally cover graphs that forbid a fixed apex graph as a minor. In another direction, F¨ uredi and Kahn [9] showed that posets with cover graphs of bounded maximum degree have dimension bounded in terms of their height.1 All this was recently generalized by Walczak [24]: Posets whose cover graphs 1We
note that the original statement of F¨ uredi and Kahn’s theorem is that posets with comparability graphs of bounded maximum degree have bounded dimension; in fact, they show a O(∆ log2 ∆) upper bound on the dimension, where ∆ denotes the maximum degree. Observe however that the comparability graph of a poset has bounded maximum degree if and only its cover graph does and its height is bounded.
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3
b4 b1
b2
b3
b4
a1
a2
a3
a4
ai < bj iff i 6= j
b3 b2 b1
a1 a2 a3 a4
Figure 1. Standard example S4 (left) and Kelly’s construction of a planar poset containing S4 (right). Recall that the standard example Sd is the poset on 2d points consisting of d minimal points a1 , . . . , ad and d maximal points b1 , . . . , bd such that ai < bj in Sd if and only if i 6= j. It is well-known and easily verified that dim(Sd ) = d. For every d > 1, Kelly’s construction provides a planar poset with 4d − 2 points containing Sd as a subposet, and hence having dimension at least d; its general definition is implicit in the figure. exclude a fixed graph as a topological minor have dimension bounded by a function of their height. (We note that Walczak’s original proof relies on the graph structure theorems for graphs excluding a fixed topological minor; see [14] for an elementary proof.) Bounded degree graphs, planar graphs, bounded treewidth graphs, and more generally graphs avoiding a fixed (topological) minor are sparse, in the sense that they have linearly many edges. This remark naturally leads one to ponder whether simply having a sparse cover graph is enough to guarantee the dimension be bounded by a function of the height. This is exactly the question we address in this work. Our main contribution is to characterize precisely the type of sparsity that is needed to ensure the property. Before explaining it let us make some preliminary observations. Clearly, asking literally that the cover graph has at most cn edges for some constant c, where n is the number of points of the poset,√is not enough since one could simply consider the union of a standard example on Θ( n) points and a large antichain. Slightly less trivially, even requiring that property to hold for every subgraph of the cover graph (that is, bounding its degeneracy) is still not enough: Define the incidence poset IG of a graph G as the height-2 poset with point set V (G) ∪ E(G), where for a vertex v and an edge e we have v < e in PG whenever v is an endpoint of e in G (and no other relation). The fact that dim(IKn ) > log log n follows easily from repeated applications of the Erd˝os-Szekeres theorem for monotone sequences (see for instance [5]). Yet, the cover graph of IKn is a clique where each edge is subdivided once, which is 2-degenerate. 1.3. Nowhere dense classes and classes with bounded expansion. Neˇsetˇril and Ossona de Mendez [17] carried out a thorough study of sparse classes of graphs over the
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last decade. Most notably, they introduced the notions of nowhere dense classes and classes with bounded expansion. Let us give some informal intuition, see Section 2 for the precise definitions. The key idea in both cases is to look at minors of bounded depth,
Bounded average degree Nowhere dense
Bounded degeneracy
Bounded expansion Almost wide
Theorem 1
Bounded queue number
Example 2
Bounded book thickness Excluded topological minor
Locally bounded treewidth k-planar Walczak [24] Excluded minor Example 2
Excluded apex minor
Bounded degree 6∆ F¨ uredi & Kahn [9]
Joret et al. [10]
Bounded genus Planar
Bounded treewidth
Streib & Trotter [21] Treewidth 6 2 + Pathwidth 6 2
dim(P ) 6 17 Bir´o et al. [1]
=
or
dim(P ) 6 1276 Joret et al. [11]
Outerplanar dim(P ) 6 4 Felsner et al. [8]
Forests dim(P ) 6 3 Trotter & Moore [23]
Figure 2. A hierarchy of sparse classes of graphs. A type of classes is drawn in grey whenever it includes a class C such that there are posets of bounded height and unbounded dimension with cover graphs in C.
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that is, minors that can be obtained by first contracting disjoint connected subgraphs of bounded radius, and then possibly removing some vertices or edges. In a nowhere dense class it is required that bounded-depth minors exclude at least some graph (which can depend on the depth). In a class with bounded expansion the requirement is stronger: For every r > 1, depth-r minors should be sparse, that is, their average degrees should be bounded by some function f (r). It is well-known that graphs with no H-minors have average degree bounded by a function of H. Thus the corresponding class of graphs has bounded expansion, since the average degree of their minors is uniformly bounded by a constant. The class of graphs with no topological H-minors also has bounded expansion, though in that case the bounding function might not be constant anymore, see [17]. These remarks show that Theorem 1 generalizes the aforementioned result for cover graphs excluding a fixed graph as a topological minor, and hence the previous body of work as well. In Figure 2 we give a summary of all the known results about posets with sparse cover graphs having their dimension bounded by a function of their height. We also mention the cases where dimension is bounded by an absolute constant, see the bottom part of the figure. Complementing our main [12] result, we describe in Section 2 a family of height-2 posets whose cover graphs form a nowhere dense class and having unbounded dimension. Thus Theorem 1 cannot be extended to nowhere dense classes. The graphs constructed moreover have locally bounded treewidth, showing that this property alone is not sufficient either. Finally, in order to complete our study of sparse cover graphs following the theory of Neˇsetˇril and Ossona de Mendez, we remark that another specialization of nowhere dense classes called almost wide classes also fail, in the sense that dimension is not bounded in terms of height for posets with cover graphs belonging to such a class. (We postpone the rather technical definition of almost wide classes until Section 2.)
1.4. Applications. Let us now turn our attention to some applications of Theorem 1 in the context of graph drawing, where natural classes with bounded expansion appear that do not fit in the previous setting of excluding a (topological) minor. Consider for instance posets whose diagrams can be drawn with ‘few’ edge crossings. If we bound the total number of crossings, then the cover graphs have bounded genus, and hence earlier results apply. On the other hand, if we only bound the number of crossings per edge in the drawing, say at most k such crossings, then we come to the well-studied class of kplanar graphs. While k-planar √ graphs are sparse—Pach and T´oth [20] proved that their average degree is at most 8 k— they do not exclude any graph as a topological minor (assuming k > 1), since every graph has a 1-planar subdivision: just start with any drawing of the graph in the plane and subdivide its edges around each crossing. Note however that this could require some edges to be subdivided many times. Indeed, one can observe that if an (< r)-subdivision of a graph G is k-planar then G is kr-planar. Combining this fact with Pach and T´oth’s result, Neˇsetˇril, Ossona de Mendez, and Wood [18] proved that the class of k-planar graphs has bounded expansion. Therefore, by Theorem 1, whenever a poset P has a k-planar cover graph, dim(P ) is bounded by a function of k and the height of P .
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Another example is given by graphs with bounded book thickness. A book embedding of a graph is a collection of half-planes (pages) all having the same line as their boundary (the spine) such that all vertices of the graph lie on the line, every edge is contained in one of the half-planes, and edges on a same page do not cross. The book thickness (also known as stack number ) of a graph is the smallest number of pages in a book embedding. Clearly, every graph on n vertices with book thickness k has at most 2kn edges. Still, every graph has a subdivision with book thickness at most 3 (see e.g. [4]). Neˇsetˇril et al. [18] observed that a result of Enomoto, Miyauchi, and Ota [6] easily implies that every class of graphs with bounded book thickness has bounded expansion. Therefore, by Theorem 1, whenever a poset P has a cover graph of book thickness at most k, its dimension is bounded by a function of k and the height of P . Yet another class of graphs with bounded expansion from graph drawing is that of graphs with bounded queue number. A queue layout of a graph is an ordering of its vertices (the spine) together with an edge coloring such that there are no two nested monochromatic edges, where two edges are nested if all four endpoints are distinct and the endpoints of one edge induce an interval on the spine containing the endpoints of the other edge. Then the queue number of a graph is the minimum number of colors in a queue layout. Every graph has a subdivision with queue number 2, as proved by Dujmovi´c and Wood [4], who also showed that graphs with bounded queue number form a class with bounded expansion. The challenging open problem in the area of queue layouts is to decide whether planar graphs have constant queue number (see e.g. [3]).
1.5. Proof ideas. We conclude this introduction with a brief overview of the two main tools used in the proof of Theorem 1. First, we use a characterization of classes with bounded expansion in terms of p-centered colorings. A p-centered coloring of a graph G is a vertex coloring of G such that, for every connected subgraph H of G, either some color is used exactly once in H, or at least p colors are used in H. Neˇsetˇril and Ossona de Mendez [15] proved that a class C has bounded expansion if and only if there is a function f such that for every integer p > 1 and every graph G ∈ C, there is a p-centered coloring of G using at most f (p) colors. Given a poset P of height h with cover graph in C, we work with a (2h − 1)-centered coloring of the cover graph with at most f (2h − 1) colors. Second, we make an exhaustive use of a decomposition for posets with large dimension called unrolling, and identify some ‘local’ part of the unrolling that still has large dimension. This simple idea, which is encapsulated in Lemma 7, is reminiscent of breadth-first search layerings for graphs and the observation that if the graph has large chromatic number then some layer still has large chromatic number. Given a poset and some (2h − 1)-centered coloring of its cover graph, we repeatedly apply the unrolling lemma as long as the dimension is large, each time winning an extra structure that is ‘floating around’ the remaining local poset with large dimension. At the very end, if enough iterations were performed (i.e. if the dimension was large enough), then we use this extra structure to find a connected subgraph of the cover graph where at most 2h − 2 colors are used and no color appears exactly once, contradicting the properties of the given (2h − 1)-centered coloring.
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We note that the idea of unrolling a poset was originally introduced by Streib and Trotter [21]. Very recently, Micek and Wiechert [14] used a similar iterative process to obtain an alternative proof of Walczak’s theorem for cover graphs in a class C excluding a fixed topological minor H: Essentially, they are able to find a subdivision of H in the cover graph when the process lasts too long. Unfortunately, there is no control on the depth of the H-minor constructed, and thus the approach of [14] cannot work directly for a class C with bounded expansion. Our proof of Theorem 1 avoids this difficulty by combining iterative unrolling with the powerful p-centered colorings. In fact, the proof is surprisingly short and simple, especially compared to previous works in the area. The paper is organized as follows. In Section 2 we give the necessary definitions for all sparse classes of graphs depicted in Figure 2, as well as the necessary background regarding posets. Along the way, we also describe constructions of posets showing that Theorem 1 cannot be extended to the classes in grey in Figure 2. We conclude that section with a key lemma about unrolling a poset with large dimension. In Section 3 we give a proof of Theorem 1.
2. Definitions and preliminaries 2.1. Posets. Let us start with basic notions about posets. All posets considered in this paper are finite. Elements of a poset P are called points. Points x, y ∈ P are said to be comparable in P if x 6 y or x > y in P . Otherwise x and y are incomparable in P . Given a poset P and a subset X of its points, the subposet of P induced by X is the poset on X where relations between the points are as in P . A set of points C ⊆ P is a chain in P if the points in C are pairwise comparable. The height of P is the maximal size of a chain in P . We write x < y in P if it holds that x 6 y and x 6= y. For distinct x, y ∈ P , if x < y in P and there is no z ∈ P with x < z < y in P then x < y is a cover relation of P . By cover(P ) we denote the cover graph of P , the (undirected) graph defined on the points of P where edges correspond to cover relations of P . A linear extension L of P is a poset on the points of P such that the points are pairwise comparable in L and x 6 y in L whenever x 6 y in P . Linear extensions L1 , . . . , Ld form a realizer of P if their intersection is equal to P , that is, x 6 y in P if and only if x 6 y in Li for each i ∈ {1, . . . , d}. The dimension of P , denoted by dim(P ), is the least number d such that there is a realizer of cardinality d of P .
2.2. Sparse graph classes. Next, we introduce the necessary definitions regarding graphs and give proper definitions for the graph classes mentioned in the introduction. All graphs in this paper are finite, simple, and undirected. Given a graph G we denote by V (G) and E(G) the vertex set and edge set of G, respectively. For a subset X ⊆ V (G) we denote by G[X] the subgraph of G induced by vertices in X. The distance between two vertices in G is the length of a shortest path between them. (Thus adjacent vertices are at distance 1; also, distance between two vertices in distinct components of G is set to +∞.) The set of all vertices at distance at most r from vertex v in G is denoted by NGr (v), and the subscript is omitted if G is clear from the context. The radius of a
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connected graph G is the least integer r > 0 for which there is a vertex v ∈ V (G) such that NGr (v) = V (G). The treewidth of a graph G, denoted by tw(G), is the least integer t > 0 for which there is a tree T and a family {T (v) | v ∈ V (G)} of non-empty subtrees of T such that |{v ∈ V (G) | x ∈ T (v)}| 6 t + 1 for each node x of T , and V (T (u)) ∩ V (T (v)) 6= ∅ for each edge uv ∈ E(G). A class of graphs C has locally bounded treewidth if there exists a function f : R → R such that tw(G[N r (v)]) 6 f (r) for every integer r > 0, graph G ∈ C and vertex v ∈ V (G). Given a partition X of the vertices of a graph G into non-empty parts inducing connected subgraphs, we denote by G/X the graph with vertex set X and edge set defined as follows: For two distinct distinct parts X, Y ∈ X , we have XY ∈ E(G/X ) if and only if there exist x ∈ X and y ∈ Y such that xy ∈ E(G). A graph H is a minor of G if H is isomorphic to a subgraph of G/X for some such partition X of V (G). A specialization of this notion is that of topological minors: H is a topological minor of a graph G if G contains a subgraph isomorphic to a subdivision of H. A class of graphs C is minor closed (topologically closed ) if every minor (topological minor, respectively) of a graph in C is also in C. We pursue with the definitions of classes with bounded expansion and nowhere dense classes. A graph H is a depth-r minor (also known as an r-shallow minor ) of a graph G if H is isomorphic to a subgraph of G/X for some partition X of V (G) into nonempty parts inducing subgraphs of radius at most r. The greatest reduced average density n (grad ) of rank r of a graph G,o denoted by ∇r (G), is defined as ∇r (G) = |E(H)| max |V | H is a depth-r minor of G . A class of graphs C has bounded expansion (H)| if there exists a function f : R → R such that ∇r (G) 6 f (r) for every integer r > 0 and graph G ∈ C. A class of graphs C is nowhere dense if for each integer r > 0 there exists a graph which is not a depth-r minor of any graph G ∈ C. It is easy to see that classes with locally bounded treewidth and classes with bounded expansion are nowhere dense. Note however that these two notions are incomparable. Two classical examples of classes with locally bounded treewidth are graphs with bounded maximum degree, and graphs excluding some apex graph A as a minor. (Recall that A is apex if A can be made planar by removing at most one vertex.) The latter is in fact a characterization of minor-closed classes with locally bounded treewidth: A minor-closed class C has locally bounded treewidth if and only if C excludes some apex graph, a fact originally proved by Eppstein [7]. Let us now define almost wide classes. For d > 1, a set of vertices X in a graph G is dindependent if every two distinct vertices in X are at distance strictly greater than d in G. A class of graphs C is almost wide if there exists an integer s > 0 such that for every integer d > 1 there is a function f : R → R such that for every integer m > 1, every graph G ∈ C of order at least f (m) contains a subset S of size at most s so that G − S has a d-independent set of size m. Neˇsetˇril and Ossona de Mendez [16, Theorem 3.23] proved that a class of graphs excluding a fixed graph as a topological minor is almost
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wide. Moreover, they proved [16, Theorem 3.13] that every hereditary class of graphs that is almost wide is also nowhere dense. (Recall that a class is hereditary if it is closed under taking induced subgraphs.) 2.3. Posets with cover graphs in a nowhere dense class. As mentioned in the introduction, the statement of Theorem 1 cannot be pushed further towards nowhere dense classes. In fact, it already fails for classes with locally bounded treewidth and hereditary classes that are almost wide, as we now show. Our construction is based on the class of graphs G with ∆(G) 6 girth(G), where ∆(G) and girth(G) denote the maximum degree and girth of G, respectively. This is a useful example of a hereditary class with locally bounded treewidth that is also almost wide but that has not bounded expansion. (Indeed, this class is invoked several times in the textbook [17].) We will use in particular that the chromatic number of these graphs is unbounded. This is a well-known fact that can be shown in multiple ways; we can note for instance that the chromatic number of the n-vertex d-regular non-bipartite Ramanujan graphs with girth √ Ωd (log n) built by Lubotzky, Phillips, and Sarnak [13] have chromatic number Ω( d) (see [13]). Proposition 2. There exists a hereditary almost wide class of graphs C with locally bounded treewidth such that posets of height 2 with cover graphs in C have unbounded dimension. Proof. For a graph G, the adjacency poset PG of G is the poset with point set {av | v ∈ V (G)} ∪ {bv | v ∈ V (G)} such that, for every two distinct vertices u, v ∈ V (G), we have au 6 bv in PG if and only if uv ∈ E(G). It is well known that dim(PG ) > χ(G). This can be seen as follows. Fix a realizer L1 , . . . , Ld of PG . For every vertex v ∈ V (G), fix a number φ(v) = i such that bv < av in Li . We claim that φ is a proper coloring of G. Consider any two adjacent vertices u and v in G and, say, φ(v) = i. Then au 6 bv < av 6 bu in Li , which witnesses that φ(u) 6= i. Therefore, dim(PG ) > χ(G). Now let C denote the class of graphs G satisfying ∆(G) 6 girth(G). As mentioned earlier, this class is hereditary, almost wide, and has locally bounded treewidth. This is not difficult to check (or see [17] for a proof). The key observation about the class C in this context is that if G ∈ C, then cover(PG ) ∈ C. This can be seen as follows: First, clearly ∆(G) = ∆(cover(PG )), so it is enough to show girth(G) 6 girth(cover(PG )). To show the latter, we remark that if C is a cycle of cover(PG ), then C naturally corresponds to a closed walk W in G of the same length. Moreover, every three consecutive vertices in that walk W are pairwise distinct, as follows from the adjacency poset construction. Hence, W contains a cycle, which is of length at most that of C. Therefore, girth(G) 6 girth(cover(PG )), as claimed. To summarize, graphs in C have unbounded chromatic number, implying that adjacency posets of these graphs have unbounded dimension. Yet, the cover graphs of these adjacency posets all belong to C, a hereditary almost wide class with locally bounded treewidth. This concludes the proof.
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2.4. Posets with cover graphs in a class with bounded expansion. There are a number of equivalent conditions for classes of graphs to have bounded expansion (see [19] for instance). As described in the introduction we make use of a characterization in terms of p-centered colorings. Recall that for a given class C with bounded expansion there is a function f such that for every integer p > 1 and every graph G ∈ C there is a p-centered coloring of G using at most f (p) colors. Therefore, the following theorem implies Theorem 1. Theorem 3. If P is a poset of height h such that its cover graph has a (2h − 1)-centered coloring using c colors, then 2 2 !3h2 (cˆ+1 h ) cˆ + 1 , dim(P ) 6 2 · 2h2 h where cˆ = max{c, 2h − 2}. The proof of Theorem 3 is given in Section 3. As a side remark, if we relax the requirement in Theorem 3 by replacing the (2h − 1)-centered coloring with a (2h − 2)-centered coloring then the statement becomes false: Proposition 4. For every h > 2 and c > 2h − 2, posets of height h whose cover graphs admit a (2h − 2)-centered coloring with at most c colors have unbounded dimension. Proof. Clearly, it is enough to show the claim for c = 2h − 2. Fix h > 2. Let Pn (n > 3) be the poset obtained from the incidence poset IKn of Kn by subdividing every comparability v < e in IKn with h − 2 distinct points; that is, points x1 , . . . , xh−2 are added such that v < x1 < · · · < xh−2 < e in Pn , and this chain displays all of their comparabilities in Pn . Note that Pn has height h. Recall that dim(IKn ) > log log(n), as discussed in the introduction. Since Pn contains IKn as an induced subposet, the dimension of Pn grows with n. Note also that the cover graph of Pn is the complete graph Kn where each edge has been subdivided exactly 2h − 3 times. Let Jn denote this graph. It remains to show that Jn has a (2h−2)-centered coloring. For this we define a coloring φ of Jn . Let V be the set of vertices of Jn with degree bigger than 2. We set φ(v) = 0 for each vertex v ∈ V , and for every two distinct vertices u, v ∈ V we let φ use each color from {1, . . . , 2h − 3} exactly once for the 2h − 3 internal vertices of the u–v path in Jn that comes from the subdivision of the edge uv. Clearly, φ uses 2h − 2 colors in total. To show that φ is (2h − 2)-centered, consider a connected subgraph H of Jn . If H avoids all vertices in V then H is a subpath of some u–v path considered earlier. In this case, all vertices in H have different colors by construction. If |V (H) ∩ V | = 1, then exactly one vertex of H is colored with 0. Finally, if |V (H) ∩ V | > 2, then H contains some u–v path as above as a subgraph, thus all 2h − 2 colors appear on H. We conclude that φ is a (2h − 2)-centered coloring. 2.5. Tools for dimension. We now give some preliminary lemmas regarding posets and their dimension. We let Inc(P ) = {(x, y) ∈ P × P | x is incomparable with y in P }
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denote the set of ordered pairs of incomparable points in P . We say that a point x ∈ P is minimal (maximal ) if there is no z ∈ P with z < x in P (x < z in P ). We denote by Min(P ) the set of minimal points in P and by Max(P ) the set of maximal points in P . The downset of a set S ⊆ P of points is defined as D(S) = {x ∈ P | ∃s ∈ S such that x 6 s in P }, and similarly we define the upset of S to be U(S) = {x ∈ P | ∃s ∈ S such that s 6 x in P }. A set I ⊆ Inc(P ) of incomparable pairs is reversible if there is a linear extension L of P that reverses each pair in I, that is y < x in L for every (x, y) ∈ I. Rephrasing the definition of dimension, dim(P ) is the least positive integer d for which there exists a partition of Inc(P ) into d reversible sets if Inc(P ) is not empty, and defined as 1 otherwise. For sets A ⊆ Min(P ) and B ⊆ Max(P ) let IncP (A, B) = {(a, b) ∈ Inc(P ) | a ∈ A and b ∈ B}. If IncP (A, B) is not empty then we define dimP (A, B) to be the least d such that IncP (A, B) can be partitioned into d reversible sets of P . Otherwise, we set dimP (A, B) = 0. We write dim(A, B) instead of dimP (A, B) when the poset is clear from the context. We use the shorthand [k] := {1, . . . , k}. Observation 5. Let P be a poset and A ⊆ Min(P ), B ⊆ Max(P ). (i) If Q is an induced subposet of P with A ⊆ Min(Q) and B ⊆ Max(Q), then dimQ (A, B) = dimP (A, B). (ii) If B1 , . . . , Bk is a partition of B, then X dimP (A, B) 6 dimP (A, Bi ), i∈[k]
and consequently there is ` ∈ [k] such that dimP (A, B` ) > dimP (A, B)/k. Proof. Item (i) follows from the observation that the restriction of any linear extension of P to the points of Q gives a linear extension of Q, and converserly, any linear extension of Q can be extended to a linear extension of P . (Note that it is crucial that Q is an induced subposet of P .) To show Item (ii), let Li (i ∈ [k]) denote the set of linear extensions of P witnessing dimP (A, Bi ).P Then L1 ∪ · · · ∪ Lk reverses all incomparable pairs in IncP (A, B) and has size at most i∈[k] dimP (A, Bi ). In our main proof we will focus on reversing incomparable pairs consisting of a minimal and a maximal point. The following observation is standard, see [11] for a proof. Observation 6. For every poset P there is a poset Q such that (i) height(Q) = height(P ), (ii) cover(Q) can be obtained from cover(P ) by attaching vertices of degree 1, and (iii) dim(P ) 6 dim(Min(Q), Max(Q)).
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G. JORET, P. MICEK, AND V. WIECHERT B1
B3
B2
A2
A1 a0
Figure 3. Unrolling of P starting from a0 . Suppose P is a connected poset, that is, that the cover graph of P is connected. Let A = Min(P ) and let B = Max(P ). Choose arbitrarily a0 ∈ A and set A0 = {a0 }. For i = 1, 2, . . . let n o [ Bi = b ∈ B − Bj | there is a ∈ Ai−1 with a 6 b in P , 16j
n o [ Ai = a ∈ A − Aj | there is b ∈ Bi with a 6 b in P . 16j
Let m be the least index such that Am is empty. Since P is connected, the sets A0 , . . . , Am−1 partition A, and the sets B1 , . . . , Bm partition B. We say that the sequence A0 , B1 , . . . , Am−1 , Bm is obtained by unrolling P from a0 . See an illustration of this decomposition in Figure 3. Also note a useful property of this construction: for every a ∈ Ai and b ∈ B with a 6 b in P we have b ∈ Bi ∪ Bi+1 , (?) for every b ∈ Bi and a ∈ A with a 6 b in P we have a ∈ Ai−1 ∪ Ai .
The following lemma intuitively says that each unrolling contains a heavy part with respect to dimension. We refer the reader to [14] for a proof. Lemma 7. Let P be a connected poset, let A = Min(P ), B = Max(P ), and suppose that dim(A, B) > 2. Consider a sequence A0 , B1 , . . . , Am−1 , Bm obtained by unrolling P . Then there is an index ` ∈ [m − 1] such that dim(A` , B` ) > dim(A, B)/2
or
dim(A` , B`+1 ) > dim(A, B)/2.
3. Proof of Theorem 3 In this section we give a proof of Theorem 3. Let thus P be a poset of height h, and let φ be a (2h − 1)-centered coloring of the cover graph of P using colors from the set {1, . . . , c}. If h = 1 then P is an antichain and dim(P ) 6 2. So we may assume h > 2. We may also suppose that c > 2h − 2, as otherwise we add artificial colors to the color set without affecting φ being (2h − 1)-centered. Thus, the inequality we need to prove is: 2 2 !3h2 (c+1 h ) c+1 dim(P ) 6 2 · 2h2 . (1) h First we apply Observation 6 to obtain a poset P 0 of height h with dim(P ) 6 dim(Min(P 0 ), Max(P 0 )) and such that cover(P 0 ) is obtained from cover(P ) by attaching
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some degree-1 vertices. We extend φ to the graph cover(P 0 ) by assigning to all new vertices a new color, namely c + 1. We claim that φ is a (2h − 1)-centered coloring of cover(P 0 ). Indeed, consider any connected subgraph H of cover(P 0 ). Since all new vertices are of degree 1, either H ∩ cover(P ) is non empty and connected, or H consists of just a single new vertex of color c + 1. Note that in the first case, either φ uses at least 2h − 1 colors on H ∩ cover(P ) (and in particular on H), or there is a color used exactly once on H ∩ cover(P ), and this color being distinct from c + 1 is also unique on H. Hence, φ is also a (2h − 1)-centered coloring of cover(P 0 ). We will focus on P 0 and show that the right-hand side of (1) is an upper bound on dim(Min(P 0 ), Max(P 0 )), which is enough since dim(P ) 6 dim(Min(P 0 ), Max(P 0 )). With a slight abuse of notation, and since we will not need to consider the old poset P anymore, for convenience we write P instead of P 0 from now on. Let us proceed with a short overview of the proof. We set up an iterative process that maintains two sets A ⊆ Min(P ) and B ⊆ Max(P ). Initially we set A := Min(P ) and B := Max(P ), and thus dim(A, B) = dim(Min(P ), Max(P )) at the beginning. We proceed with a new iteration as long as the current sets A and B satisfy dim(A, B) > 3. An iteration consists in finding non-empty subsets A0 ⊆ A and B 0 ⊆ B by unrolling the current poset (this will be explained shortly) that have some useful structural properties, and satisfying −2 dim(A, B) c+1 0 0 dim(A , B ) > . (2) · h 2 h (Note that, since dim(A0 , B 0 ) is an integer, we always have dim(A0 , B 0 ) > 1.) Then, at the end of the iteration, we update the sets A and B by setting A := A0 and B := B 0 . Every iteration has a corresponding type, a 3-tuple (α, X, Y ) where α ∈ {1, 2, 3} encodes the unrolling type (to be defined later), and X and Y are non-empty subsets of the color 2 set {1, . . . , c + 1}, each of size at most h. Thus, there are at most 3h2 c+1 different h iteration types (since c > 2h − 2). A key property of our approach will be that no two iterations can have the same type; this will be shown at the end of the proof. Note that 2 this implies that after at most 3h2 c+1 iterations we must have dim(A, B) 6 2, which h gives the desired bound on dim(Min(P ), Max(P )). We continue with a precise description of an iteration and the definition of its type. Suppose we are given A ⊆ Min(P ) and B ⊆ Max(P ) with dim(A, B) > 3. First, we clean the picture by ruling out some trivialities. It is well-known that the dimension of a poset is witnessed by the dimension of a subposet induced by a single component of the cover graph. (To be precise, this holds except when the poset is a union of chains.) In our setting, we are interested in bounding dim(A, B). Using that dim(A, B) > 3, we can find sets A00 ⊆ A and B 00 ⊆ B such that • dim(A00 , B 00 ) = dim(A, B); • U(A00 ) ∩ D(B 00 ) induces a connected subgraph of cover(P ), and • A00 ⊆ D(B 00 ) and B 00 ⊆ U(A00 ).
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(We leave the easy proof to the reader.) Define Q to be the subposet of P induced by U(A00 ) ∩ D(B 00 ). Thus, Q is connected, A00 = Min(Q), and B 00 = Max(Q). By Observation 5(i), we have dimQ (A00 , B 00 ) = dimP (A00 , B 00 ). Choose a point a0 ∈ A00 arbitrarily, and let A0 , B1 , A1 , . . . , Am−1 , Bm be the sequence obtained by unrolling Q from a0 . By Lemma 7 there is an index ` ∈ [m − 1] such that dimQ (A` , B` ) > dimQ (A, B)/2 or dimQ (A` , B`+1 ) > dimQ (A, B)/2. Since Q is an induced subposet of P , this implies dimP (A` , B` ) > dimP (A, B)/2
or
dimP (A` , B`+1 ) > dimP (A, B)/2.
These two possibilities are responsible for the first coordinate of an iteration type, the unrolling type: If dimP (A` , B` ) > dimP (A, B)/2 then the unrolling type is 1 or 2, depending on whether ` = 1 or ` > 2. Otherwise, we have dimP (A` , B`+1 ) > dimP (A, B)/2, and the unrolling type is 3. We now consider each of the three unrolling types separately. Unrolling type 1. Suppose that the unrolling type is 1, thus dimP (A1 , B1 ) > dimP (A, B)/2. Observe that a0 < b in Q for all points b ∈ B1 in this case. Suppose that the current iteration is the i-th iteration in our process. For every point b ∈ B1 , we fix a chain C i (b) = {y1 , . . . , yr } of Q with y1 = a0 and yr = b, and such that yk < yk+1 is a cover relation in P for each k ∈ [r−1]. The chain has length at most h as this is the height of P . Thus, the set of colors φ(C i (b)) is a non-empty subset of [c + 1] of size at most h, and hence there are at most h c+1 possible values h i of φ(C (b)) (as c > 2h − 2). These values induce a partition of B1 into at most h c+1 h sets. Using Observation 5(ii), it follows that for some Y ⊆ [c + 1] of size at most h we −1 have that B 0 := {b ∈ B1 | φ(C i (b)) = Y } satisfies dimP (A1 , B 0 ) > dimP2(A,B) · h c+1 . h Note that points of A1 that are below all the points from B 0 in P do not contribute to dimP (A1 , B 0 ) as they are not involved in any pair in IncP (A1 , B 0 ). We set A0 := A1 − {a ∈ A1 | a 6 b in P for all b ∈ B 0 }, and thus have −1 dimP (A, B) c+1 0 0 0 dimP (A , B ) = dimP (A1 , B ) > · h . 2 h Clearly, A0 and B 0 are non empty sets since otherwise we would have dim(A0 , B 0 ) = 0, while the right-hand side of the inequality is strictly greater than 0. We define the iteration type in this case to be (1, ∅, Y ), and note that the desired lower bound on dimP (A0 , B 0 ) (c.f. (2)) is satisfied. Let us summarize some important properties that we have in this case: (1.1) for every b ∈ B 0 , the points of C i (b) form a chain with b = max(C i (b));
SPARSITY AND DIMENSION
(1.2) (1.3) (1.4) (1.5)
15
for every b ∈ B 0 , the chain C i (b) induces a connected subgraph of cover(P ); for every b ∈ B 0 , we have φ(C i (b)) = Y ; there exists a ∈ A − A0 such that min(C i (b)) = a for every b ∈ B 0 , and for every a ∈ A0 , there exists b ∈ B 0 such that a and b are incomparable in P .
Unrolling type 2. Suppose next that the unrolling type is 2, so dimP (A` , B` ) > dimP (A, B)/2 for some ` > 2. In particular, the sets A`−1 and B`−1 exist. Recall that, by the definition of unrolling Q, every point in B` is above some point in A`−1 in Q, and also every point in A`−1 is below some point in B`−1 in Q. For every point b ∈ B` we define the following sets: First, we fix a chain b = y1 > · · · > yj = a of Q such that a ∈ A`−1 and yk > yk+1 is a cover relation in P for every k ∈ [j − 1]. Let r ∈ [j − 1] be minimal such that yr ∈ D(B`−1 ), which is well defined since a = yj ∈ D(B`−1 ). (The two downsets are with respect to poset Q.) Next, we fix a chain yr = x1 < x2 < · · · < xs in Q which is such that xs ∈ B`−1 and xk < xk+1 is a cover relation in P for every k ∈ [s − 1]. Now we wish to associate two chains to the point b. For this suppose that we are currently in the i-th iteration. Then let Li (b) = {x1 , . . . , xs } and Ri (b) = {y1 , . . . , yr−1 }. Since Li (b) and Ri (b) form chains of 2 size at most h and h − 1 in P , respectively, there at most h c+1 possibilities for the h i i pair (φ(L (b)), φ(R (b))). Considering the value of the pair (φ(Li (b)), φ(Ri (b))) for each point b ∈ B` , we obtain a 2 partition of B` into at most h c+1 sets. As a consequence, there are sets X, Y ⊆ [c+1] h of size at most h and h − 1, respectively, such that B 0 := {b ∈ B` | φ(Li (b)) = X and φ(Ri (b)) = Y } satisfies −2 dimP (A, B) c+1 0 dimP (A` , B ) > · h . 2 h We set A0 := A` and define the iteration type to be (2, X, Y ). Note again that A0 and B 0 are non empty, and that inequality (2) is satisfied. It is straightforward to verify the following properties: For every point b ∈ B 0 , points in Li (b) and Ri (b) form chains in P such that (2.1) (2.2) (2.3) (2.4) (2.5) (2.6)
max(Li (b)) ∈ B − B 0 and max(Ri (b)) = b; Li (b) ∪ Ri (b) induces a connected subgraph of cover(P ); the upset of Ri (b) in P avoids points of Li (b); the upset of A0 in P avoids points of Li (b) (by the property (?) of unrolling); the upset of A in P contains Li (b), and φ(Li (b)) = X and φ(Ri (b)) = Y , and |X| 6 h and |Y | 6 h − 1.
Unrolling type 3. Assume now that the unrolling type is 3. Then dim(A` , B`+1 ) > dim(A, B)/2 for some ` > 2. Note that ` > 1 since dim(A0 , B1 ) = 0. In particular, the sets B` and A`−1 exist. We will proceed analogously to the previous case.
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For every a ∈ A` we define the following sets: First, we fix a chain a = y1 < y2 < · · · < yr = b of Q such that b ∈ B` and yk < yk+1 is a cover relation in P for every k ∈ [r − 1]. Let j ∈ [r − 1] be minimal such that yj ∈ U(A`−1 ). Next, we fix a chain x1 < x2 < · · · < xs = yj of Q with x1 ∈ A`−1 and with xk < xk+1 being a cover relation in P for every k ∈ [s − 1]. Suppose we are in the i-th iteration. Then we set Li (a) = {x1 , . . . , xs } and Ri (a) = {y1 , . . . , yj−1 }. 2 As before, for a point a ∈ A` , there are at most h c+1 possible evaluations of the h 2 i i pair (φ(L (a)), φ(R (a))). This induces a partition of A` into at most h c+1 sets. We h deduce in turn that there are sets X, Y ⊆ [c+1] of size at most h and h−1, respectively, such that A0 := {a ∈ A` | φ(Li (a)) = X and φ(Ri (a)) = Y } satisfies −2 dim(A, B) c+1 0 dim(A , B`+1 ) > · h . 2 h We set B 0 := B`+1 and define the iteration type to be (3, X, Y ). Clearly, A0 and B 0 are non empty, and inequality (2) is satisfied. This concludes our study of the three unrolling types. Iteration types do not reoccur. Now we will prove that no two iterations have the same type. We begin with iterations that have unrolling types 1. Claim. For each Y ⊆ [c + 1], there is at most one iteration with type (1, ∅, Y ). Proof. Suppose for a contradiction that two different iterations have the same type (1, ∅, Y ), say the i-th and j-th iterations, with i < j. Let Ai , B i and Aj , B j denote the value of the subsets A, B at the end iteration i and j, respectively. Then Aj ⊆ Ai and B j ⊆ B i . Let ai and aj be points witnessing property (1.4) for iteration i and j respectively (notice that in each case it suffices to take the point from which the current poset Q was unrolled). By construction dim(Aj , B j ) > 1, and hence B j is not empty. Choose arbitrarily a point b ∈ B j ⊆ B i and let b0 ∈ B i be such that (aj , b0 ) ∈ Inc(Ai , B i ) (such a b0 exists by (1.5)). Now consider the three chains C i (b0 ), C i (b), and C j (b); see Figure 4 (left) for an illustration. Since C i (b0 ) and C i (b) both contain ai by (1.4), and C i (b) and C j (b) both contain b, it follows by (1.2) that the union U of the three chains induce a connected subgraph of cover(P ). As iterations i and j both have type (1, ∅, Y ), by (1.3) we have φ(C i (b0 )) = φ(C i (b)) = φ(C j (b0 )) = Y , and hence at most h colors are used on U . Furthermore, each color in Y is used at least twice on U . This is because the chains C i (b0 ) and C j (b) must be disjoint, since otherwise aj and b0 would be comparable in P . However, since h < 2h − 1, this contradicts the fact that φ is a (2h − 1)-centered coloring of cover(P ). Claim. For each X, Y ⊆ [c + 1] there is at most one iteration of type (2, X, Y ). Proof. Suppose that there are two different iterations of type (2, X, Y ). Say this is the case for the i-th and j-th iterations, with i < j. Let Ai , B i and Aj , B j denote the
SPARSITY AND DIMENSION b0
Bi
ai
b
Bi
Bj
aj
Ai
Aj
b0
Li (b0 ) Ri (b0 )
17 Bj
Ai
b
Aj
Figure 4. Left: union of the three chains C i (b0 ), C i (b), and C j (b). Right: union of the four chains Li (b0 ), Ri (b0 ), Lj (b), and Rj (b). Dotted lines indicate incomparabilities. value of the subsets A, B at the end iteration i and j, respectively. Then Aj ⊆ Ai and B j ⊆ B i . By construction dim(Aj , B j ) > 1, and hence B j is not empty. Fix a point b ∈ B j and consider the two chains Lj (b) and Rj (b). Let b0 = max(Lj (b)), which is contained in set B i by (2.1). For the rest of the argument we focus on the four chains Li (b0 ), Ri (b0 ), Lj (b), Rj (b), and their union U = Li (b0 ) ∪ Ri (b0 ) ∪ Lj (b) ∪ Rj (b); see Figure 4 (right) for an illustration. Since iterations i and j share type (2, X, Y ), by (2.6) we have X = φ(Li (b0 )) = φ(Lj (b)), Y = φ(Ri (b0 )) = φ(Rj (b)), and hence φ(U ) = X ∪ Y . Observe that U induces a connected subgraph of cover(P ). Indeed, unions Li (b0 ) ∪ Ri (b0 ) and Lj (b) ∪ Rj (b) are connected in cover(P ) by (2.2), and both unions share point b0 . Furthermore, since b0 is contained in Lj (b) and Ri (b0 ), we conclude that color sets X and Y are not disjoint. Together with (2.6) this implies |X ∪ Y | 6 2h − 2. Now note that Li (b0 ) and Lj (b) cannot intersect since the upset of Ai in P is disjoint from Li (b0 ) (by (2.4) for iteration i), but it contains Lj (b) (by (2.5) for iteration j). Hence no color from X is unique on U . Chains Ri (b0 ) and Rj (b) cannot intersect as well since otherwise b0 = max(Ri (b0 )) would be in the upset of Rj (b). However, this contradicts (2.3) since b0 is contained in Lj (b). It follows that no color from Y is unique on U either. This contradicts the fact that φ is (2h − 1)-centered. It remains to consider the case of two different iterations sharing a type that is of the form (3, X, Y ). However, the argument goes along similar lines as for the type (2, X, Y ). Therefore, we omit a detailed treatment here. This conludes the proof of Theorem 3.
Acknowledgements The first author thanks David R. Wood for asking whether Theorem 1 holds in the case of cover graphs with bounded queue number, which prompted this line of research.
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