Dominated colorings of graphs - Semantic Scholar
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Dominated colorings of graphs Houcine Boumediene Merouane · Mohammed Haddad? · Mustapha Chellali · Hamamache Kheddouci
Received: date / Accepted: date
Abstract In this paper, we introduce and study a new coloring problem of a graph called the dominated coloring. A dominated coloring of a graph G is a proper vertex coloring of G such that each color class is dominated by at least one vertex of G. The minimum number of colors among all dominated colorings is called the dominated chromatic number, denoted by χdom (G). In this paper, we establish the close relationship between the dominated chromatic number χdom (G) and the total domination number γt (G); and the equivalence for triangle-free graphs. We study the complexity of the problem by proving its NP-completeness for arbitrary graphs having χdom (G) ≥ 4 and by giving a polynomial time algorithm for recognizing graphs having χdom (G) ≤ 3. We also give some bounds for planar and star-free graphs and exact values for split graphs. Keywords dominated coloring · total domination · algorithms · triangle-free graphs · star-free graphs · split graphs Mathematics Subject Classification (2000) 05C15 · 05C85
? Contact author information: [email protected], Universit´ e Claude Bernard Lyon 1, Campus de la doua, Bat. Nautibus, 43 Bd du 11 Novembre 1918, F-69622, Villeurbanne, France. Tel: +33 423 26 44 65, Fax: +33 472 43 15 37
This research was supported by ”Region Rhˆ one-Alpes COOPERA project” and ”Programmes Nationaux de Recherche: Code 8/u09/510”. H. Boumediene Merouane, M. Chellali LAMDA-RO Laboratory, Department of Mathematics University of Blida. B.P. 270, Blida, Algeria. M. Haddad, H. Kheddouci Laboratoire LIRIS, UMR CNRS 5205, Universit´ e de Lyon F-69003 Universit´ e Claude Bernard Lyon 1, 43, Bd du 11 novembre 1918 F-69622 Villeurbanne Cedex, France
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1 Introduction Let G = (V, E) be a graph such that V is the vertex set and E is the edge set. A subset S ⊆ V is a dominating set of G if every vertex in V −S has a neighbor in S and is a total dominating set, if every vertex in V has a neighbor in S. The domination number γ(G) (respectively, total domination number γt (G)) is the minimum cardinality of a dominating set (respectively, total dominating set) of G. For a comprehensive survey of domination in graphs and its variations, see [11, 12]. A proper vertex coloring of a graph is a vertex coloring such that no two adjacent vertices have the same color. A proper coloring c using k colors is called a (proper) k-coloring. A subset of vertices colored with the same color is called a color class and every such class forms an independent set. Thus, finding a proper k-coloring of a graph G is equivalent to the partitioning of its vertex set into k independent sets C1 , C2 , . . . , Ck . The minimum number of colors among all proper colorings of G is the chromatic number of G, denoted by χ(G). A graph admitting a proper k-coloring is said to be k-colorable, and it is said to be k-chromatic if its chromatic number is exactly k. There exist plenty of variants of classical graph coloring [6, 14]. Moreover, graph coloring and domination problems are often in relation. Chellali and Volkmann [5] showed some relations between the chromatic number and some domination parameters in the graph. More recently, the dominator coloring problem was defined. Indeed, Gera et al. have introduced in 2006 the dominator coloring [9] as a proper coloring such that every vertex has to dominate at least one color class (possibly its own class). The minimum number of colors among all dominator colorings of G is the dominator chromatic number of G, denoted by χd (G). Gera studied further the problem in [7, 8], she proved its NP-completeness and showed that every graph G satisfies max{γ(G), χ(G)} ≤ χd (G) ≤ γ(G) + χ(G). Chellali and Maffray gave a polynomial-time algorithm for computing χd on P4 -free graphs [4]. Quite recently, Arumugam et al. showed that the dominator coloring problem is NPhard on bipartite, planar and split graphs [2]. Further results on this coloring could be found in [3, 4, 9, 7, 8]. A slightly different coloring, called the strict strong coloring, was introduced by Haddad and Kheddouci [10]. In that coloring, each vertex has to dominate a color class different from its own class. It was shown that computing the minimum cardinality of a strict strong coloring of an arbitrary graph is an NP-hard problem, but it can be done in polynomial-time on trees. For the problems mentioned above, the domination property is defined on the vertices. Indeed, it is required that each vertex dominates a color class, so the color classes are not necessarily all dominated. In contrast, in the problem we consider, each color class must be dominated and the vertices are not necessarily all dominating vertices. We define a new coloring that we call the dominated coloring as follows: a proper vertex coloring of G is a dominated coloring if and only if every color class is dominated by at least one vertex of G. More formally, a k-dominated coloring of G is a proper k-coloring
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{C1 , C2 , . . . , Ck } of G where for every index i ∈ {1, 2, . . . , k}, there exists a vertex u ∈ V such that Ci ⊆ N (u) (i.e. u is adjacent to every vertex of Ci ). We say that u dominates Ci or Ci is dominated by u. The vertex u is then a dominating vertex. The dominated chromatic number χdom (G) is the minimum number of colors among all dominated colorings of G. G is said to be k-dominated chromatic if its dominated chromatic number is exactly k. This paper is organized as follows: We first give some introductory results in Section 2. In Section 3, we propose existence and complexity results. Then, in Section 4, we study the dominated coloring of triangle-free, planar, star-free and split graphs. Finally we give, in Section 5, a polynomial-time algorithm for recognizing graphs G having χdom (G) ≤ 3. 2 Terminology and Preliminaries Before presenting our results, we need to introduce some additional but standard notations and definitions. Given a graph G = (V, E) and X ⊆ V , we denote by G[X] the subgraph of G induced by X. A component of a graph G is an induced subgraph of G that is connected and maximal with this property. The maximum degree of a graph G is denoted by ∆(G). The open and closed neighborhood of a vertex x ∈ V are denoted by N (x) and N [x], respectively. We denote by Pn the path on n vertices and by Cn the cycle on n vertices. The complete graph on n vertices is denoted by Kn and the complete graph of order 3 is called a triangle. The complete bipartite graph with parts of orders r and s is denoted by Kr,s . The star is the complete bipartite graph K1,k denoted by Sk with k ≥ 1. A bi-star Bp,q is a graph formed by two stars Sp and Sq by adding an edge between their center vertices. Given any graph F , a graph G is F-free if it does not have any induced subgraph isomorphic to F . A tree is any connected graph that contains no cycle. Let us start with some observations. Let G = (V, E) be a graph. The dominated coloring is defined for graphs without isolated vertices. In the rest of the paper, we consider only such graphs. 1. χdom (G) ≥ 2. Moreover, χdom (G) = 2 if and only if G is a bi-star to which we can add some edges not inducing any triangle. 2. If G is disconnected, then χdom (G) = Σi χdom (Gi ), where Gi is the ith connected component of G. This result holds since a color of a component cannot appear elsewhere. 3. χdom (G) ≥ γt (G). Indeed, consider a minimum dominated coloring of G. Let us construct Dt , a total dominating set of G, as follows: from each color class, take one of its dominating vertices. The set Dt is a dominating set of cardinality χdom (G). Moreover, Dt is a total dominating set since each of its vertices has a color and is therefore dominated by some other vertex of Dt . We note that the difference χdom (G) − γt (G) can be arbitrarily large which may be seen by considering complete graphs of higher
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order. However, we will see in Section 4 that equality between χdom (G) and γt (G) holds for a large class of graphs. 4. If G is a triangle-free graph, then χdom (G) ≤ 2γ(G). Indeed, let D be a minimum dominating set of G. Color every vertex of D with a new color. Since the set of neighbors of every vertex of D is an independent set, a second new color is given for each neighborhood. Thus, χdom (G) ≤ 2|D|. Note that the equality between χdom (G) and 2γ(G) even holds for trees. This can be seen by the tree Tk formed by a path Pk on k vertices, where for each vertex v of Pk , two new vertices v 0 and v 00 are added with edges vv 0 and v 0 v 00 . One can see easily that χdom (G) = 2k = 2γ(G).
3 Existence and complexity results We begin this section by giving some results that deal with the existence of graphs with prescribed values χdom , χ and γ. We recall that the k th power of a graph G, denoted by Gk , is the graph obtained from G by adding edges between all pairs of vertices separated by distance in G of at most k. Theorem 1 If a and b are two integers with a ≥ b ≥ 2, then there exists a graph G with dominated chromatic number χdom (G) = a and chromatic number χ(G) = b. Proof Let a and b be two positive integers such that a ≥ b ≥ 2. Suppose a = b. Then consider G to be a complete multipartite graph with b partitions. Clearly, χ(G) = b. Moreover, since in every coloring of G with b colors, there exists a vertex which dominates each color class (partition), the proper coloring of that graph is also a dominated coloring. Hence, χdom (G) = a = b. Now, let us suppose a > b. We construct a particular graph G such that χdom (G) = a and χ(G) = b. This construction is based on the (b − 1)st power of a path. Consider a path Pn with n ≥ b. Then, for its power Pnb−1 , we have χ(Pnb−1 ) = b. On the other hand, if we give interest to the dominated coloring of Pnb−1 , we observe that a color could not appear on more than two vertices (see Figure 1). 2b vertices with b colors
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Fig. 1 Power of path with a = 6 and b = 4
In fact, if we try to color three or more vertices with the same color, either the coloring wouldn’t remain proper or the color cannot be dominated. In both
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cases, the obtained coloring is not a dominated coloring. Hence, b new colors will be necessary and sufficient to color 2b consecutive vertices. Now, we have to determine the value n for which a colors will be necessary and sufficient to t u color all vertices of G. It is enough to consider n = 2bb ab c + a mod b. We say that we attach a vertex to another one if we join them by an edge. Theorem 2 If a and b are two integers with a ≥ b ≥ 2, then there exists a graph G with dominated chromatic number χdom (G) = a and domination number γ(G) = b. Proof Let a and b be two positive integers such that a ≥ b ≥ 2. Consider the graph obtained from the complete graph Ka by attaching b new vertices to b vertices from Ka . Easily, one can check that χdom (G) = a and γ(G) = b. For example, see the graph in Figure 2.
Fig. 2 Graph with a = 5 and b = 3
t u The second part of this section focuses on the complexity study of the problem. Hence, we give interest to the decision problem whether an arbitrary graph admits a dominated coloring with at most k colors. We give the following formalization: k-dominated Coloring Problem: Instance: a graph G = (V, E) without isolated vertices and a positive integer k. Question: is there a dominated coloring of G with at most k colors? Theorem 3 For k ≥ 4, the k-dominated coloring problem is NP-Complete. Proof Since verifying if a coloring is a dominated coloring could be performed in polynomial time, the k-dominated coloring problem is in NP. Now, we give a polynomial time reduction from a k-coloring problem which is known to be NP-Complete, for k ≥ 3. We construct a graph G0 from G by adding a
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u
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G Fig. 3 The graphs G and
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dominating vertex u to G (that is, u is adjacent to all vertices of G). We show that G admits a proper coloring with k colors if and only if G0 admits a dominated coloring with k + 1 colors. First, we prove the necessity. Let C = {C1 , C2 , . . . , Ck } be a proper kcoloring of G. We construct a (k + 1)-dominated coloring D of G0 such that: D = {D1 = C1 , D2 = C2 , . . . , Dk = Ck , Dk+1 = {u}}. It is easy to see that D is a dominated coloring of G0 because: – D is proper. – each color class other than u’s is dominated by u and the color class containing u is dominated by any other vertex. Now, we prove the sufficiency. Let D = {C1 , C2 , . . . , Ck , Ck+1 } be a (k +1)dominated coloring of G0 . We construct a proper k-coloring C of G. Since D is dominated, D is also proper. So, there exists a color class Ci such that Ci = {u}. Then C is the set consisting of the k other color classes Cj , j 6= i. t u Instances of the k-dominated coloring problem having k ≤ 3 will be studied in Section 5.
4 χdom for some graph classes In this section, we study the dominated coloring problem on triangle-free, star-free and split graphs. Recall from Section 2, that the total domination number γt is a lower bound for the dominated chromatic number of arbitrary graphs. Here, we show that γt is also an upper bound for χdom (G) when G is a triangle-free graph. Proposition 1 Let G be a triangle-free graph. Then χdom (G) ≤ γt (G). Proof Consider a minimum total dominating set Dt of G. Construct pairs of adjacent dominating vertices (some vertices would stay single). For every
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obtained pair (a, b) we give a new color cb to b and the neighborhood of a, and give another new color ca to a and the neighborhood of b (the subgraph induced by the two classes of colors ca and cb forms a spanning subgraph isomorphic to a bistar centered at a and b). On the other hand, a single dominating vertex c is necessarily adjacent to a unique dominating vertex belonging to the pair (a, b). Hence, the vertex c is already colored with either ca or cb . We then give a new color cc to the neighborhood of c belonging to V \ Dt . For illustration, see Figure 4. ca
cb
ca a cb
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Fig. 4 Coloring of a, b, c and their neighborhoods
We finally obtain a dominated coloring of G with at most γt (G) colors. t u As a consequence of 3. in Section 2 and Proposition 1, we obtain the following result:
Theorem 4 The dominated chromatic number of a triangle-free graph G is equal to its total domination number.
Since there exists a linear time algorithm for finding a minimum total dominating set in trees due to Laskar et al. [13], the previous result gives us a linear-time algorithm for finding a minimum dominated coloring of trees. We obtain:
Corollary 1 There exists a linear time algorithm that gives a minimum dominated coloring of any tree.
Proof The algorithm given in [13] computes an optimal total dominating set for any tree in linear time. Since the coloring operations given by the proof of Proposition 1 are performed once for each vertex, the coloring phase is also linear. t u We saw in Section 2, that the ratio χdom is bounded from above by 2 for γ triangle-free graphs. We show below that it is bounded by χ for every graph.
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Proposition 2 Let G be a graph without isolated vertices. Then χdom (G) ≤ χ(G) · γ(G).
Proof Consider a graph G and a minimum dominating set D of G. We obtain a dominated coloring of G by giving distinct colors to each vertex v of D and at most χ(G) − 1 new colors to the vertices of N (v). Hence, we use at most (χ(G) − 1) γ(G) + γ(G) = γ(G)χ(G) colors, and so χdom (G) ≤ γ(G)χ(G). t u The bound of Proposition 2 is tight for complete graphs. Recall from [1] that every planar graph is 4-chromatic, hence the following corollary is straightforward: Corollary 2 Let G be a planar graph without isolated vertices. Then χdom (G) ≤ 4 · γ(G). The bound of Corollary 2 is tight for the class of planar graphs formed from k copies of the complete planar graph of order 4, denoted by K41 , K42 , .., K4k , by attaching three new vertices, a, b and c, to the same vertex chosen arbitrarily from each complete graph. And by adding an edge between the vertices c and b attached to Ki and Ki+1 , respectively, with i = 1, .., k − 1. For illustration, see Figure 5.
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Fig. 5 Planar graphs with χdom = 4γ .
Next we give interest to the class of star-free graphs. Proposition 3 Let G be a graph of order n. If G is Sk -free, where k ≥ 2, n then we have χdom (G) ≥ k−1 . Proof Consider a minimum dominated coloring of G. Any color class would not have more than k − 1 vertices; otherwise, a vertex dominating such color class will induce a star of order at least k, a contradiction. t u
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It is clear that the bound is attained for complete graphs. Note that complete graphs are S2 -free graphs. Next, we give for k ≥ 3 a characterization of (Sk , K3 )-free graphs (star- and triangle-free) for which the bound is reached. Proposition 4 Let k ≥ 3 be an integer and G a (Sk , K3 )-free graph of order n if and only if G is obtained from one or more bisn. Then χdom (G) = k−1 tars Bk−2,k−2 by adding edges (possibly none) between leaves so that G is an (Sk , K3 )-free graph. Proof Assume that G is formed from bistars Bk−2,k−2 by adding edges (possibly none) between leaves so that G is an (Sk , K3 )-free graph. We construct a dominated coloring of G by giving two new colors to each bistar (a color is given to a center and leaves of the other center). Since each bistar has 2k − 2 n n colors and so χdom (G) ≤ k−1 . The equality is obtained vertices, we use 2 2k−2 from Proposition 3. n Conversely, let G be a (Sk , K3 )-free graph with χdom (G) = k−1 , and consider a minimum dominated coloring {V1 , V2 , ...Vχdom }. Since G is Sk -free, |Vi | ≤ k − 1 for every i. X Now if for some j, |Vj | < k − 1, then n = |Vi | < (k − 1)χdom = n, a contradiction. Hence each color class contains exactly k − 1 vertices. Now, assume that the color class Vi is dominated by vj ∈ Vj . Clearly i 6= j. If any vertex x not in Vi is adjacent to vj , then Vi ∪ {x, vj } may induce an Sk and to avoid this, x must be adjacent to a vertex z in Vi , but then {x, vj , z} induces a K3 , a contradiction. Hence, all neighbors of vj are in Vi , in particular the vertex dominating Vj , say vi . Likewise no vertex of V \Vj is adjacent to vi . It follows that the subgraph induced by Vi ∪ Vj contains a spanning graph isomorphic to a bistar Bk−2,k−2 , all neighbors of vi and vj belong to Vi ∪Vj , and each dominating vertex has exactly degree k−1. The same argument to that used with Vi and Vj can be applied for the remaining color classes. Implying that χdom (G) is even and G is formed from bistars Bk−2,k−2 , where possible further edges are between leaves without common centers. Therefore, vertices that are not dominating and do not belong to the same color class may be adjacent, so G is as described in the statement of the proposition. t u Since every graph is an S∆+1 -free graph, we have the following corollaries to Propositions 3 and 4, respectively. Corollary 3 Let G be a graph with maximum degree ∆ and order n. Then we n . have χdom (G) ≥ ∆
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Corollary 4 Let G be a K3 -free graph with maximum degree ∆ and order n n. Then χdom (G) = ∆ if and only if G is formed from bistars B∆−1,∆−1 by adding edges (possibly none) between leaves so that the obtained graph is a K3 -free with maximum degree ∆. In the following, we give interest to split graphs. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Theorem 5 Let G be a split graph such that its maximum clique is of order k. Then χdom (G) = χ(G) = ω(G) = k. Proof Consider a minimum dominated coloring of G. Obviously, χdom (G) ≥ k. We give now a construction that yields a dominated coloring of any split graph with k colors. 1 2
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j Fig. 6 Circular order
First, we give to each vertex of the clique one unique new color from the set {1, . . . , k}. We arrange the clique vertices according to a circular order function defined on the set {1, . . . , k} as follows: j ∈ {1, . . . , k} ⇒ next(j) = j mod (k) + 1. We now color the vertices of the independent set of the split graph G. Let i be a vertex of the independent set and let N (i) be the set (of colors) of its neighbors. The color of i is given by the following formula: c(i) = min{j : j ∈ {{1, . . . , k} r N (i)} ∧ next(j) ∈ N (i)}. Claim 1: Every vertex is properly colored. A vertex of the independent set cannot be adjacent to all vertices of the clique (otherwise the graph G would have a clique with more than k = ω(G) vertices). So, at least one color from the set {1, . . . , k} would be available for that vertex. Hence, for each vertex i from the independent set, the resulting color set {{1, . . . , k} r N (i)} is not empty.
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Claim 2: Every color class is dominated. From the given construction, one can observe that each color j will appear only in the neighborhood of the vertex from the clique colored with the color next(j). By the two previous claims, we obtain that the proposed construction gives a dominated coloring for every split graph with k colors. t u
5 Recognition of 3-dominated colorable graphs Recall from Section 3 that the k-dominated coloring problem is NP-complete, for k ≥ 4. In this section, we are interested in the following decision problem related to instances where k ≤ 3: 3-dominated coloring problem: Instance: a graph G = (V, E) without isolated vertices. Question: is there a dominated coloring of G with at most 3 colors? We recall from Section 2, that a graph G is 2-dominated chromatic if and only if G is obtained from a bistar by adding edges (possibly none) between leaves of different centers. We shall characterize the class of 3-dominated chromatic graphs. It should be mentioned that there is no forbidden subgraphs characterization of 3-dominated chromatic graphs. Indeed, the graph G in Figure 7 has dominated chromatic number equal to 4. However, the graph G0 obtained from G by adding a new vertex x and edges xa, xb and xc, has dominated chromatic number equal to 3. x
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b Fig. 7 The graphs G with χdom (G) = 4 and G0 with χdom (G0 ) = 3
The following proposition gives a trivial constructive characterization of graphs G with χdom (G) ≤ 3: Proposition 5 Let G be a graph. Then, χdom (G) ≤ 3 if and only if G is obtained from three non-empty independent sets Vi , 1 ≤ i ≤ 3, as follows: I) For each Vi , 1 ≤ i ≤ 3:
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I.1) Choose an arbitrary vertex xi ∈ Vj ∪ Vk with j = i + 1 (mod 3) and k = i + 2 (mod 3); I.2) Add edges between xi and every vertex of Vi . II) Add edges (possibly none) between vertices from different sets. The last characterization highlights the class of 3-dominated chromatic graphs. On the one hand, two colors may be reciprocally dominated, i.e. each having at least one dominating vertex colored with the second color. We call such a coloring an edge-3-dominated coloring. On the other hand, there may exist three dominating vertices of the three colors inducing a triangle. We call such a coloring a triangle-3-dominated coloring. Observe that both situations can occur simultaneously. For illustration, see Figure 8. 2 1 2
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Fig. 8 3-dominated coloring: a) edge-3-dominated coloring; b) triangle-3-dominated coloring; c) edge- and triangle-3-dominated coloring;
We give a polynomial-time algorithm for the 3-dominated coloring problem. First, our approach searches for a 2-dominated coloring or an edge-3dominated coloring of the graph G. If such colorings do not exist, the algorithm searches for a triangle-3-dominated coloring of G. The algorithm for the 3-dominated coloring problem: The output of the algorithm is either a minimum dominated coloring using at most three colors, 1, 2 and 3, or a negative answer to the problem 3-dominated coloring. Recall from 1. in Section 2 that a 2-dominated coloring like an edge-3-dominated coloring induces the existence of a particular edge connecting the centers of the bi-star. Thus, in every iteration of the first loop (resp. the second one), we consider a new edge e (resp. triangle K3 ) and search for a 2-dominated coloring or an edge-3-dominated coloring (resp. triangle-3dominated coloring) based on e (resp. on K3 ), if not we jump to the next edge (resp. triangle). Before we give more details, we shall introduce additional notations. For a subset A of V , let h(A) denote the number of connected components of G[A] and for two vertices x, y in the same component of G[A], we denote by dA (x, y) the length of a shortest path between x and y in G[A]. Moreover, for x ∈ A,
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we denote by bicol(x, a, b, A) the process whose output is a partial coloring c0 , as follows: for every y ∈ A such that x and y belong to the same connected component in G[A], if dA (x, y) is even then c0 (y) = a else c0 (y) = b. Note that c0 may not be proper. Algorithm 1 Algorithm for the 3-dominated coloring problem Require: A connected graph G without isolated vertices. Ensure: Determines if G admits a dominated coloring with at most 3 colors. If yes, gives a minimum dominated coloring of G. for all e ∈ E(G) do Call Procedure A (Algorithm 2) if G admits a 2-dominated coloring based on e or an edge-3-dominated coloring based on e then Exit with answer χdom (G) ≤ 3. end if end for for all induced subgraphs K3 of G do Call Procedure B (Algorithm 3) if G admits a triangle-3-dominated coloring, based on the induced K3 then Exit with answer χdom (G) ≤ 3. end if end for Exit with answer χdom (G) > 3.
Next, we give the description of procedure A. Procedure A: The details of Procedure A are given in Algorithm 2. The procedure contains six steps and requires a connected graph G = (V, E) and an edge e = uv of G. In the following, we shall analyse each step. Without loss of generality, at the first step, at most three colors, say 1, 2 and 3, are sufficient and necessary to color u, v and their common neighborhood, respectively. In Step 2, since u and v are presumed to dominate the classes of colors 2 and 1 respectively, each vertex external to the neighborhoods of u and v has to receive the color 3. Also the neighbors of u (resp. of v) not adjacent to any other neighbor of u (resp. v) may receive the color 2 (resp. 1), with no effect on the optimality of the coloring. In Step 3, we extend the partial coloring using the trivial fact that if two different colors are present in the open neighborhood of some uncolored vertex, then this vertex must receive the third color. Let Nu and Nv be the current sets of non-colored vertices adjacent to u and v respectively. Note that Nu and Nv are updated throughout the algorithm. In Step 4, we form the set CAN of candidate vertices to dominate the class of color 3. Clearly, if there are vertices already colored with color 3, CAN should be the common neighborhood of these vertices. Otherwise, the color 3 should be present in every component of G[Nu ] and G[Nv ]. Hence, G[Nu ] or G[Nv ] has to be connected in order to be able to contain a dominating vertex of the
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class of color 3. Then, if there is no candidate vertex to dominate the class of color 3 (CAN = ∅), either G does not admit an edge-3-dominated coloring based on the edge uv, or G is a 2-dominated chromatic graph i.e. G is a bi-star. In Step 5, we verify whether there exists a vertex of CAN able to dominate the class of color 3. Let x be a vertex of CAN . In Step 5.1, since x is presumed to dominate the class of color 3, we extend the coloring such that x and every vertex not adjacent to x takes a color other than 3. In Step 5.2, all non-colored vertices are adjacent to x, some of them may be adjacent to vertices colored with color 3. We extend the coloring for these last ones. In Step 5.3, the remaining non-colored vertices are adjacent to x and not adjacent to vertices colored with color 3. Thus, an arbitrary coloring does not affect the optimality of final coloring. In Step 5.4, we verify whether the obtained coloring is a dominated coloring, otherwise we continue with another vertex of CAN . Finally, if Step 6 is reached, then G does not admit an edge-3-dominated coloring based on the edge uv.
Algorithm 2 Procedure A Require: A connected graph G = (V, E), an edge e = uv of G. Ensure: The procedure determines whether G admits a 2-dominated coloring or an edge3-dominated coloring based on e. If yes, it gives a minimum dominated coloring of G. Throughout the algorithm, let Nu and Nv be the current sets of non-colored vertices adjacent to u and v, respectively. STEP 1 Give the colors 1 and 2 to u and v, respectively. Let COM = N (u) ∩ N (v), give the color 3 to all vertices of COM . STEP 2 Give the color 2 (resp. the color 1) to the isolated vertices in G[Nu ] (resp. G[Nv ]). Let EXT = V \ (N [u] ∪ N [v]); give the color 3 to all vertices of EXT . STEP 3 for all x ∈ EXT ∪ COM do bicol(x, 3, 2, {x} ∪ Nu ). bicol(x, 3, 1, {x} ∪ Nv ). end for STEP 4 //Forming CAN , the set of candidate vertices to dominate the class of //color 3. CAN = ∅. STEP 4.1 //A dominating vertex of the class of color 3 has to be adjacent to all //vertices already colored with 3 if such vertices exist, otherwise, //to all components of G[Nu ] and G[Nv ]. if EXT ∪ COM 6= ∅ then CAN = {x ∈ V : ∀y ∈ EXT ∪ COM : xy ∈ E} else if h(Nv ) ≥ 2 and h(Nu ) = 1 then CAN = Nu . end if
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if h(Nu ) ≥ 2 and h(Nv ) = 1 then CAN = Nv . end if if h(Nu ) ≤ 1 and h(Nv ) ≤ 1 then CAN = Nu ∪ Nv . end if end if STEP 4.2 if CAN = ∅ then if COM ∪ EXT ∪ Nu ∪ Nv 6= ∅ then Return FALSE. //G does not admit an edge-3-dominated coloring //based on e = uv. else Return TRUE. //G is a 2-dominated chromatic graph. end if end if STEP 5 //Search for a vertex x from CAN to dominate the class of color 3. for all x ∈ CAN do STEP 5.1 //A unique possible extension of the coloring such that x and every //non-adjacent vertex to x receives a color other than 3. for all y ∈ ({x} ∪ (Nu ∪ Nv ) \ N (x)) do if y ∈ Nu then bicol(y, 2, 3, Nu ). end if if y ∈ Nv then bicol(y, 1, 3, Nv ). end if end for STEP 5.2 //A unique possible extension of the coloring such that every non-colored //vertex is adjacent to x and not adjacent to vertices colored with color 3. Let Ax be the component of G[N (u)], or of G[N (v)], containing x. Let A3x be the subset of vertices of Ax colored with 3. for all y ∈ (Nu ∪ Nv ) ∩ N (A3x ) do if y ∈ Nu then bicol(y, 2, 3, Nu ). end if if y ∈ Nv then bicol(y, 1, 3, Nv ). end if end for STEP 5.3 //All non-colored vertices are adjacent to x and non-adjacent to all //vertices colored with 3. We give them an arbitrary proper color. for all y ∈ Nu ∪ Nv do if y ∈ Nu then bicol(y, 2, 3, Nu ). end if if y ∈ Nv then bicol(y, 1, 3, Nv ). end if end for STEP 5.4 //At this step all vertices are colored. if the obtained coloring is proper and x dominate the class of color 3 then Return TRUE. //G admits an edge-3-dominated coloring. end if end for STEP 6 Return FALSE.//G does not admit neither a 2-dominated coloring nor an //edge-3-dominated coloring based on e = uv.
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In the following, we give the description of procedure B.
Procedure B: The procedure B consists of two steps and checks whether G admits a triangle-3-dominated coloring based on an induced triangle K = {x1 , x2 , x3 } of G. For each permutation (i, j, k) of the set {1, 2, 3}, we denote by Pi , the private neighborhood of xi in G with respect to K, i.e. Pi = N (xi ) \ (N (xj ) ∪ N (xk )); and by CNi,j , the common neighborhood of xi and xj except xk , i.e. CNi,j = (N (xi ) ∩ N (xj )) \ {xk }. In the first step, we construct the unique possible coloring. In Step 2, the non-colored vertices belong to the private neighborhood sets. Then, we check whether the obtained coloring is a triangle-3-dominated coloring, clockwise if possible or anti-clockwise, otherwise.
Algorithm 3 Procedure B Require: A connected graph G = (V, E), an induced triangle K = {x1 , x2 , x3 } of G. Ensure: The procedure determines whether G admits a triangle-3-dominated coloring based on K. If yes, it gives a minimum dominated coloring of G. STEP 1 Give the colors 1, 2 and 3 to the vertices x1 , x2 and x3 respectively. for all permutations (i, j, k) of {1, 2, 3} do Give the color k to the vertices of CNi,j . end for STEP 2 Give the color i + 1 (mod 3) to the vertices of Pi , with i : 1..3. if the coloring is proper then Return TRUE. else Give the color i − 1 (mod 3) to the vertices of Pi , with i : 1..3. if the coloring is proper then Return TRUE. end if end if Return FALSE.
Complexity analysis: Let ∆ be the maximum degree and m be the number of edges of G. In the procedure A, the process bicol applied on a component of G[Nu ] or G[Nv ] is in O(∆). Testing whether a coloring is proper and if x dominates the color 3 are done in O(∆2 ) and O(∆), respectively. In the procedure B, O(∆2 ) times are spent for computing the private and the common neighborhoods. The procedure A will be executed O(m) times (corresponding to the complexity of browsing all edges of the graph). The procedure B will be executed O(m · ∆2 ) times (corresponding to the complexity of finding all triangles of the graph). Hence, we need at most O(m ·
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∆2 ) and O(m · ∆4 ) for the first loop and the second loop of Algorithm for 3dominated coloring problem (Algorithm 1), respectively. Hence, the algorithm 3-dominated coloring recognition is polynomial.
Acknowledgments We thank the referees for comments that helped us to improve the paper.
References 1. K. Appel, W. Haken, Every planar map is four colorable, Bulletin of the American Mathematical Society, 82(5), pp. 711-712, 1976. 2. S. Arumugam, K. Raja Chandrasekar, N. Misra, P. Geevarghese, S. Saurabh, Algorithmic aspects of dominator colorings in graphs, IWOCA’2011, LNCS 7056, pp. 19-30, 2011. 3. H. Boumediene Merouane and M. Chellali, On the dominator colorings in trees, Discussiones Mathematicae Graph Theory, 32(4), pp. 677-683, 2012. 4. M. Chellali, F. Maffray, Dominator colorings in some classes of graphs, Graphs and Combinatorics, 28, pp. 97-107, 2012. 5. M. Chellali and L. Volkmann, Relations between the lower domination parameters and the chromatic number of a graph, Discrete Mathematics, 274(1-3), pp.1-8, 2004. 6. G. Chen, A. Gyarfas and R. Schelp, Vertex colorings with a distance restriction, Discrete Mathematics, 191(1-3), pp. 65-82, 1998. 7. R. Gera, On the dominator colorings in bipartite graphs, Information Technology: New Generations, pp. 947-952, 2007. 8. R. Gera, On dominator colorings in graphs, Graph Theory Notes of New York LII, pp. 25-30, 2007. 9. R. Gera, S. Horton and C. Rasmussen, Dominator colorings and safe clique partitions, Congressus Numerantium, 181, pp. 19-32, 2006. 10. M. Haddad and H. Kheddouci. A strict strong coloring of trees, Information Processing Letters, 109(18), pp. 1047-1054, 2009. 11. T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998. 12. T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics, Marcel Dekker, New York, 1998. 13. R. Laskar, J.Pfaff , S. M. Hedetniemi and S. T. Hedetniemi, On the algorithmic complexity of total domination, SIAM Journal on Algebraic and Discrete Methods, 5(3), pp. 420-425, 1984. 14. E. Malaguti and P. Toth. A survey on vertex coloring problems, International Transactions in Operational Research, 17: 1-34, 2010. 15. J. Pfaff, R.C. Laskar, S.T. Hedetniemi, NP-completeness of total and connected domination and irredundance for bipartite graphs, Technical Report 428, Clemson University, Dept. Math. Sciences, 1983.
Dominated colorings of graphs Houcine Boumediene Merouane · Mohammed Haddad? · Mustapha Chellali · Hamamache Kheddouci
Received: date / Accepted: date
Abstract In this paper, we introduce and study a new coloring problem of a graph called the dominated coloring. A dominated coloring of a graph G is a proper vertex coloring of G such that each color class is dominated by at least one vertex of G. The minimum number of colors among all dominated colorings is called the dominated chromatic number, denoted by χdom (G). In this paper, we establish the close relationship between the dominated chromatic number χdom (G) and the total domination number γt (G); and the equivalence for triangle-free graphs. We study the complexity of the problem by proving its NP-completeness for arbitrary graphs having χdom (G) ≥ 4 and by giving a polynomial time algorithm for recognizing graphs having χdom (G) ≤ 3. We also give some bounds for planar and star-free graphs and exact values for split graphs. Keywords dominated coloring · total domination · algorithms · triangle-free graphs · star-free graphs · split graphs Mathematics Subject Classification (2000) 05C15 · 05C85
? Contact author information: [email protected], Universit´ e Claude Bernard Lyon 1, Campus de la doua, Bat. Nautibus, 43 Bd du 11 Novembre 1918, F-69622, Villeurbanne, France. Tel: +33 423 26 44 65, Fax: +33 472 43 15 37
This research was supported by ”Region Rhˆ one-Alpes COOPERA project” and ”Programmes Nationaux de Recherche: Code 8/u09/510”. H. Boumediene Merouane, M. Chellali LAMDA-RO Laboratory, Department of Mathematics University of Blida. B.P. 270, Blida, Algeria. M. Haddad, H. Kheddouci Laboratoire LIRIS, UMR CNRS 5205, Universit´ e de Lyon F-69003 Universit´ e Claude Bernard Lyon 1, 43, Bd du 11 novembre 1918 F-69622 Villeurbanne Cedex, France
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1 Introduction Let G = (V, E) be a graph such that V is the vertex set and E is the edge set. A subset S ⊆ V is a dominating set of G if every vertex in V −S has a neighbor in S and is a total dominating set, if every vertex in V has a neighbor in S. The domination number γ(G) (respectively, total domination number γt (G)) is the minimum cardinality of a dominating set (respectively, total dominating set) of G. For a comprehensive survey of domination in graphs and its variations, see [11, 12]. A proper vertex coloring of a graph is a vertex coloring such that no two adjacent vertices have the same color. A proper coloring c using k colors is called a (proper) k-coloring. A subset of vertices colored with the same color is called a color class and every such class forms an independent set. Thus, finding a proper k-coloring of a graph G is equivalent to the partitioning of its vertex set into k independent sets C1 , C2 , . . . , Ck . The minimum number of colors among all proper colorings of G is the chromatic number of G, denoted by χ(G). A graph admitting a proper k-coloring is said to be k-colorable, and it is said to be k-chromatic if its chromatic number is exactly k. There exist plenty of variants of classical graph coloring [6, 14]. Moreover, graph coloring and domination problems are often in relation. Chellali and Volkmann [5] showed some relations between the chromatic number and some domination parameters in the graph. More recently, the dominator coloring problem was defined. Indeed, Gera et al. have introduced in 2006 the dominator coloring [9] as a proper coloring such that every vertex has to dominate at least one color class (possibly its own class). The minimum number of colors among all dominator colorings of G is the dominator chromatic number of G, denoted by χd (G). Gera studied further the problem in [7, 8], she proved its NP-completeness and showed that every graph G satisfies max{γ(G), χ(G)} ≤ χd (G) ≤ γ(G) + χ(G). Chellali and Maffray gave a polynomial-time algorithm for computing χd on P4 -free graphs [4]. Quite recently, Arumugam et al. showed that the dominator coloring problem is NPhard on bipartite, planar and split graphs [2]. Further results on this coloring could be found in [3, 4, 9, 7, 8]. A slightly different coloring, called the strict strong coloring, was introduced by Haddad and Kheddouci [10]. In that coloring, each vertex has to dominate a color class different from its own class. It was shown that computing the minimum cardinality of a strict strong coloring of an arbitrary graph is an NP-hard problem, but it can be done in polynomial-time on trees. For the problems mentioned above, the domination property is defined on the vertices. Indeed, it is required that each vertex dominates a color class, so the color classes are not necessarily all dominated. In contrast, in the problem we consider, each color class must be dominated and the vertices are not necessarily all dominating vertices. We define a new coloring that we call the dominated coloring as follows: a proper vertex coloring of G is a dominated coloring if and only if every color class is dominated by at least one vertex of G. More formally, a k-dominated coloring of G is a proper k-coloring
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{C1 , C2 , . . . , Ck } of G where for every index i ∈ {1, 2, . . . , k}, there exists a vertex u ∈ V such that Ci ⊆ N (u) (i.e. u is adjacent to every vertex of Ci ). We say that u dominates Ci or Ci is dominated by u. The vertex u is then a dominating vertex. The dominated chromatic number χdom (G) is the minimum number of colors among all dominated colorings of G. G is said to be k-dominated chromatic if its dominated chromatic number is exactly k. This paper is organized as follows: We first give some introductory results in Section 2. In Section 3, we propose existence and complexity results. Then, in Section 4, we study the dominated coloring of triangle-free, planar, star-free and split graphs. Finally we give, in Section 5, a polynomial-time algorithm for recognizing graphs G having χdom (G) ≤ 3. 2 Terminology and Preliminaries Before presenting our results, we need to introduce some additional but standard notations and definitions. Given a graph G = (V, E) and X ⊆ V , we denote by G[X] the subgraph of G induced by X. A component of a graph G is an induced subgraph of G that is connected and maximal with this property. The maximum degree of a graph G is denoted by ∆(G). The open and closed neighborhood of a vertex x ∈ V are denoted by N (x) and N [x], respectively. We denote by Pn the path on n vertices and by Cn the cycle on n vertices. The complete graph on n vertices is denoted by Kn and the complete graph of order 3 is called a triangle. The complete bipartite graph with parts of orders r and s is denoted by Kr,s . The star is the complete bipartite graph K1,k denoted by Sk with k ≥ 1. A bi-star Bp,q is a graph formed by two stars Sp and Sq by adding an edge between their center vertices. Given any graph F , a graph G is F-free if it does not have any induced subgraph isomorphic to F . A tree is any connected graph that contains no cycle. Let us start with some observations. Let G = (V, E) be a graph. The dominated coloring is defined for graphs without isolated vertices. In the rest of the paper, we consider only such graphs. 1. χdom (G) ≥ 2. Moreover, χdom (G) = 2 if and only if G is a bi-star to which we can add some edges not inducing any triangle. 2. If G is disconnected, then χdom (G) = Σi χdom (Gi ), where Gi is the ith connected component of G. This result holds since a color of a component cannot appear elsewhere. 3. χdom (G) ≥ γt (G). Indeed, consider a minimum dominated coloring of G. Let us construct Dt , a total dominating set of G, as follows: from each color class, take one of its dominating vertices. The set Dt is a dominating set of cardinality χdom (G). Moreover, Dt is a total dominating set since each of its vertices has a color and is therefore dominated by some other vertex of Dt . We note that the difference χdom (G) − γt (G) can be arbitrarily large which may be seen by considering complete graphs of higher
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order. However, we will see in Section 4 that equality between χdom (G) and γt (G) holds for a large class of graphs. 4. If G is a triangle-free graph, then χdom (G) ≤ 2γ(G). Indeed, let D be a minimum dominating set of G. Color every vertex of D with a new color. Since the set of neighbors of every vertex of D is an independent set, a second new color is given for each neighborhood. Thus, χdom (G) ≤ 2|D|. Note that the equality between χdom (G) and 2γ(G) even holds for trees. This can be seen by the tree Tk formed by a path Pk on k vertices, where for each vertex v of Pk , two new vertices v 0 and v 00 are added with edges vv 0 and v 0 v 00 . One can see easily that χdom (G) = 2k = 2γ(G).
3 Existence and complexity results We begin this section by giving some results that deal with the existence of graphs with prescribed values χdom , χ and γ. We recall that the k th power of a graph G, denoted by Gk , is the graph obtained from G by adding edges between all pairs of vertices separated by distance in G of at most k. Theorem 1 If a and b are two integers with a ≥ b ≥ 2, then there exists a graph G with dominated chromatic number χdom (G) = a and chromatic number χ(G) = b. Proof Let a and b be two positive integers such that a ≥ b ≥ 2. Suppose a = b. Then consider G to be a complete multipartite graph with b partitions. Clearly, χ(G) = b. Moreover, since in every coloring of G with b colors, there exists a vertex which dominates each color class (partition), the proper coloring of that graph is also a dominated coloring. Hence, χdom (G) = a = b. Now, let us suppose a > b. We construct a particular graph G such that χdom (G) = a and χ(G) = b. This construction is based on the (b − 1)st power of a path. Consider a path Pn with n ≥ b. Then, for its power Pnb−1 , we have χ(Pnb−1 ) = b. On the other hand, if we give interest to the dominated coloring of Pnb−1 , we observe that a color could not appear on more than two vertices (see Figure 1). 2b vertices with b colors
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Fig. 1 Power of path with a = 6 and b = 4
In fact, if we try to color three or more vertices with the same color, either the coloring wouldn’t remain proper or the color cannot be dominated. In both
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cases, the obtained coloring is not a dominated coloring. Hence, b new colors will be necessary and sufficient to color 2b consecutive vertices. Now, we have to determine the value n for which a colors will be necessary and sufficient to t u color all vertices of G. It is enough to consider n = 2bb ab c + a mod b. We say that we attach a vertex to another one if we join them by an edge. Theorem 2 If a and b are two integers with a ≥ b ≥ 2, then there exists a graph G with dominated chromatic number χdom (G) = a and domination number γ(G) = b. Proof Let a and b be two positive integers such that a ≥ b ≥ 2. Consider the graph obtained from the complete graph Ka by attaching b new vertices to b vertices from Ka . Easily, one can check that χdom (G) = a and γ(G) = b. For example, see the graph in Figure 2.
Fig. 2 Graph with a = 5 and b = 3
t u The second part of this section focuses on the complexity study of the problem. Hence, we give interest to the decision problem whether an arbitrary graph admits a dominated coloring with at most k colors. We give the following formalization: k-dominated Coloring Problem: Instance: a graph G = (V, E) without isolated vertices and a positive integer k. Question: is there a dominated coloring of G with at most k colors? Theorem 3 For k ≥ 4, the k-dominated coloring problem is NP-Complete. Proof Since verifying if a coloring is a dominated coloring could be performed in polynomial time, the k-dominated coloring problem is in NP. Now, we give a polynomial time reduction from a k-coloring problem which is known to be NP-Complete, for k ≥ 3. We construct a graph G0 from G by adding a
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u
G’
G Fig. 3 The graphs G and
G0
dominating vertex u to G (that is, u is adjacent to all vertices of G). We show that G admits a proper coloring with k colors if and only if G0 admits a dominated coloring with k + 1 colors. First, we prove the necessity. Let C = {C1 , C2 , . . . , Ck } be a proper kcoloring of G. We construct a (k + 1)-dominated coloring D of G0 such that: D = {D1 = C1 , D2 = C2 , . . . , Dk = Ck , Dk+1 = {u}}. It is easy to see that D is a dominated coloring of G0 because: – D is proper. – each color class other than u’s is dominated by u and the color class containing u is dominated by any other vertex. Now, we prove the sufficiency. Let D = {C1 , C2 , . . . , Ck , Ck+1 } be a (k +1)dominated coloring of G0 . We construct a proper k-coloring C of G. Since D is dominated, D is also proper. So, there exists a color class Ci such that Ci = {u}. Then C is the set consisting of the k other color classes Cj , j 6= i. t u Instances of the k-dominated coloring problem having k ≤ 3 will be studied in Section 5.
4 χdom for some graph classes In this section, we study the dominated coloring problem on triangle-free, star-free and split graphs. Recall from Section 2, that the total domination number γt is a lower bound for the dominated chromatic number of arbitrary graphs. Here, we show that γt is also an upper bound for χdom (G) when G is a triangle-free graph. Proposition 1 Let G be a triangle-free graph. Then χdom (G) ≤ γt (G). Proof Consider a minimum total dominating set Dt of G. Construct pairs of adjacent dominating vertices (some vertices would stay single). For every
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obtained pair (a, b) we give a new color cb to b and the neighborhood of a, and give another new color ca to a and the neighborhood of b (the subgraph induced by the two classes of colors ca and cb forms a spanning subgraph isomorphic to a bistar centered at a and b). On the other hand, a single dominating vertex c is necessarily adjacent to a unique dominating vertex belonging to the pair (a, b). Hence, the vertex c is already colored with either ca or cb . We then give a new color cc to the neighborhood of c belonging to V \ Dt . For illustration, see Figure 4. ca
cb
ca a cb
cb b ca
cc
ca c cc
Fig. 4 Coloring of a, b, c and their neighborhoods
We finally obtain a dominated coloring of G with at most γt (G) colors. t u As a consequence of 3. in Section 2 and Proposition 1, we obtain the following result:
Theorem 4 The dominated chromatic number of a triangle-free graph G is equal to its total domination number.
Since there exists a linear time algorithm for finding a minimum total dominating set in trees due to Laskar et al. [13], the previous result gives us a linear-time algorithm for finding a minimum dominated coloring of trees. We obtain:
Corollary 1 There exists a linear time algorithm that gives a minimum dominated coloring of any tree.
Proof The algorithm given in [13] computes an optimal total dominating set for any tree in linear time. Since the coloring operations given by the proof of Proposition 1 are performed once for each vertex, the coloring phase is also linear. t u We saw in Section 2, that the ratio χdom is bounded from above by 2 for γ triangle-free graphs. We show below that it is bounded by χ for every graph.
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Proposition 2 Let G be a graph without isolated vertices. Then χdom (G) ≤ χ(G) · γ(G).
Proof Consider a graph G and a minimum dominating set D of G. We obtain a dominated coloring of G by giving distinct colors to each vertex v of D and at most χ(G) − 1 new colors to the vertices of N (v). Hence, we use at most (χ(G) − 1) γ(G) + γ(G) = γ(G)χ(G) colors, and so χdom (G) ≤ γ(G)χ(G). t u The bound of Proposition 2 is tight for complete graphs. Recall from [1] that every planar graph is 4-chromatic, hence the following corollary is straightforward: Corollary 2 Let G be a planar graph without isolated vertices. Then χdom (G) ≤ 4 · γ(G). The bound of Corollary 2 is tight for the class of planar graphs formed from k copies of the complete planar graph of order 4, denoted by K41 , K42 , .., K4k , by attaching three new vertices, a, b and c, to the same vertex chosen arbitrarily from each complete graph. And by adding an edge between the vertices c and b attached to Ki and Ki+1 , respectively, with i = 1, .., k − 1. For illustration, see Figure 5.
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Fig. 5 Planar graphs with χdom = 4γ .
Next we give interest to the class of star-free graphs. Proposition 3 Let G be a graph of order n. If G is Sk -free, where k ≥ 2, n then we have χdom (G) ≥ k−1 . Proof Consider a minimum dominated coloring of G. Any color class would not have more than k − 1 vertices; otherwise, a vertex dominating such color class will induce a star of order at least k, a contradiction. t u
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It is clear that the bound is attained for complete graphs. Note that complete graphs are S2 -free graphs. Next, we give for k ≥ 3 a characterization of (Sk , K3 )-free graphs (star- and triangle-free) for which the bound is reached. Proposition 4 Let k ≥ 3 be an integer and G a (Sk , K3 )-free graph of order n if and only if G is obtained from one or more bisn. Then χdom (G) = k−1 tars Bk−2,k−2 by adding edges (possibly none) between leaves so that G is an (Sk , K3 )-free graph. Proof Assume that G is formed from bistars Bk−2,k−2 by adding edges (possibly none) between leaves so that G is an (Sk , K3 )-free graph. We construct a dominated coloring of G by giving two new colors to each bistar (a color is given to a center and leaves of the other center). Since each bistar has 2k − 2 n n colors and so χdom (G) ≤ k−1 . The equality is obtained vertices, we use 2 2k−2 from Proposition 3. n Conversely, let G be a (Sk , K3 )-free graph with χdom (G) = k−1 , and consider a minimum dominated coloring {V1 , V2 , ...Vχdom }. Since G is Sk -free, |Vi | ≤ k − 1 for every i. X Now if for some j, |Vj | < k − 1, then n = |Vi | < (k − 1)χdom = n, a contradiction. Hence each color class contains exactly k − 1 vertices. Now, assume that the color class Vi is dominated by vj ∈ Vj . Clearly i 6= j. If any vertex x not in Vi is adjacent to vj , then Vi ∪ {x, vj } may induce an Sk and to avoid this, x must be adjacent to a vertex z in Vi , but then {x, vj , z} induces a K3 , a contradiction. Hence, all neighbors of vj are in Vi , in particular the vertex dominating Vj , say vi . Likewise no vertex of V \Vj is adjacent to vi . It follows that the subgraph induced by Vi ∪ Vj contains a spanning graph isomorphic to a bistar Bk−2,k−2 , all neighbors of vi and vj belong to Vi ∪Vj , and each dominating vertex has exactly degree k−1. The same argument to that used with Vi and Vj can be applied for the remaining color classes. Implying that χdom (G) is even and G is formed from bistars Bk−2,k−2 , where possible further edges are between leaves without common centers. Therefore, vertices that are not dominating and do not belong to the same color class may be adjacent, so G is as described in the statement of the proposition. t u Since every graph is an S∆+1 -free graph, we have the following corollaries to Propositions 3 and 4, respectively. Corollary 3 Let G be a graph with maximum degree ∆ and order n. Then we n . have χdom (G) ≥ ∆
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Corollary 4 Let G be a K3 -free graph with maximum degree ∆ and order n n. Then χdom (G) = ∆ if and only if G is formed from bistars B∆−1,∆−1 by adding edges (possibly none) between leaves so that the obtained graph is a K3 -free with maximum degree ∆. In the following, we give interest to split graphs. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Theorem 5 Let G be a split graph such that its maximum clique is of order k. Then χdom (G) = χ(G) = ω(G) = k. Proof Consider a minimum dominated coloring of G. Obviously, χdom (G) ≥ k. We give now a construction that yields a dominated coloring of any split graph with k colors. 1 2
k Next
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k−1
j Fig. 6 Circular order
First, we give to each vertex of the clique one unique new color from the set {1, . . . , k}. We arrange the clique vertices according to a circular order function defined on the set {1, . . . , k} as follows: j ∈ {1, . . . , k} ⇒ next(j) = j mod (k) + 1. We now color the vertices of the independent set of the split graph G. Let i be a vertex of the independent set and let N (i) be the set (of colors) of its neighbors. The color of i is given by the following formula: c(i) = min{j : j ∈ {{1, . . . , k} r N (i)} ∧ next(j) ∈ N (i)}. Claim 1: Every vertex is properly colored. A vertex of the independent set cannot be adjacent to all vertices of the clique (otherwise the graph G would have a clique with more than k = ω(G) vertices). So, at least one color from the set {1, . . . , k} would be available for that vertex. Hence, for each vertex i from the independent set, the resulting color set {{1, . . . , k} r N (i)} is not empty.
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Claim 2: Every color class is dominated. From the given construction, one can observe that each color j will appear only in the neighborhood of the vertex from the clique colored with the color next(j). By the two previous claims, we obtain that the proposed construction gives a dominated coloring for every split graph with k colors. t u
5 Recognition of 3-dominated colorable graphs Recall from Section 3 that the k-dominated coloring problem is NP-complete, for k ≥ 4. In this section, we are interested in the following decision problem related to instances where k ≤ 3: 3-dominated coloring problem: Instance: a graph G = (V, E) without isolated vertices. Question: is there a dominated coloring of G with at most 3 colors? We recall from Section 2, that a graph G is 2-dominated chromatic if and only if G is obtained from a bistar by adding edges (possibly none) between leaves of different centers. We shall characterize the class of 3-dominated chromatic graphs. It should be mentioned that there is no forbidden subgraphs characterization of 3-dominated chromatic graphs. Indeed, the graph G in Figure 7 has dominated chromatic number equal to 4. However, the graph G0 obtained from G by adding a new vertex x and edges xa, xb and xc, has dominated chromatic number equal to 3. x
a
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b Fig. 7 The graphs G with χdom (G) = 4 and G0 with χdom (G0 ) = 3
The following proposition gives a trivial constructive characterization of graphs G with χdom (G) ≤ 3: Proposition 5 Let G be a graph. Then, χdom (G) ≤ 3 if and only if G is obtained from three non-empty independent sets Vi , 1 ≤ i ≤ 3, as follows: I) For each Vi , 1 ≤ i ≤ 3:
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I.1) Choose an arbitrary vertex xi ∈ Vj ∪ Vk with j = i + 1 (mod 3) and k = i + 2 (mod 3); I.2) Add edges between xi and every vertex of Vi . II) Add edges (possibly none) between vertices from different sets. The last characterization highlights the class of 3-dominated chromatic graphs. On the one hand, two colors may be reciprocally dominated, i.e. each having at least one dominating vertex colored with the second color. We call such a coloring an edge-3-dominated coloring. On the other hand, there may exist three dominating vertices of the three colors inducing a triangle. We call such a coloring a triangle-3-dominated coloring. Observe that both situations can occur simultaneously. For illustration, see Figure 8. 2 1 2
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(b)
3 (c)
Fig. 8 3-dominated coloring: a) edge-3-dominated coloring; b) triangle-3-dominated coloring; c) edge- and triangle-3-dominated coloring;
We give a polynomial-time algorithm for the 3-dominated coloring problem. First, our approach searches for a 2-dominated coloring or an edge-3dominated coloring of the graph G. If such colorings do not exist, the algorithm searches for a triangle-3-dominated coloring of G. The algorithm for the 3-dominated coloring problem: The output of the algorithm is either a minimum dominated coloring using at most three colors, 1, 2 and 3, or a negative answer to the problem 3-dominated coloring. Recall from 1. in Section 2 that a 2-dominated coloring like an edge-3-dominated coloring induces the existence of a particular edge connecting the centers of the bi-star. Thus, in every iteration of the first loop (resp. the second one), we consider a new edge e (resp. triangle K3 ) and search for a 2-dominated coloring or an edge-3-dominated coloring (resp. triangle-3dominated coloring) based on e (resp. on K3 ), if not we jump to the next edge (resp. triangle). Before we give more details, we shall introduce additional notations. For a subset A of V , let h(A) denote the number of connected components of G[A] and for two vertices x, y in the same component of G[A], we denote by dA (x, y) the length of a shortest path between x and y in G[A]. Moreover, for x ∈ A,
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we denote by bicol(x, a, b, A) the process whose output is a partial coloring c0 , as follows: for every y ∈ A such that x and y belong to the same connected component in G[A], if dA (x, y) is even then c0 (y) = a else c0 (y) = b. Note that c0 may not be proper. Algorithm 1 Algorithm for the 3-dominated coloring problem Require: A connected graph G without isolated vertices. Ensure: Determines if G admits a dominated coloring with at most 3 colors. If yes, gives a minimum dominated coloring of G. for all e ∈ E(G) do Call Procedure A (Algorithm 2) if G admits a 2-dominated coloring based on e or an edge-3-dominated coloring based on e then Exit with answer χdom (G) ≤ 3. end if end for for all induced subgraphs K3 of G do Call Procedure B (Algorithm 3) if G admits a triangle-3-dominated coloring, based on the induced K3 then Exit with answer χdom (G) ≤ 3. end if end for Exit with answer χdom (G) > 3.
Next, we give the description of procedure A. Procedure A: The details of Procedure A are given in Algorithm 2. The procedure contains six steps and requires a connected graph G = (V, E) and an edge e = uv of G. In the following, we shall analyse each step. Without loss of generality, at the first step, at most three colors, say 1, 2 and 3, are sufficient and necessary to color u, v and their common neighborhood, respectively. In Step 2, since u and v are presumed to dominate the classes of colors 2 and 1 respectively, each vertex external to the neighborhoods of u and v has to receive the color 3. Also the neighbors of u (resp. of v) not adjacent to any other neighbor of u (resp. v) may receive the color 2 (resp. 1), with no effect on the optimality of the coloring. In Step 3, we extend the partial coloring using the trivial fact that if two different colors are present in the open neighborhood of some uncolored vertex, then this vertex must receive the third color. Let Nu and Nv be the current sets of non-colored vertices adjacent to u and v respectively. Note that Nu and Nv are updated throughout the algorithm. In Step 4, we form the set CAN of candidate vertices to dominate the class of color 3. Clearly, if there are vertices already colored with color 3, CAN should be the common neighborhood of these vertices. Otherwise, the color 3 should be present in every component of G[Nu ] and G[Nv ]. Hence, G[Nu ] or G[Nv ] has to be connected in order to be able to contain a dominating vertex of the
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class of color 3. Then, if there is no candidate vertex to dominate the class of color 3 (CAN = ∅), either G does not admit an edge-3-dominated coloring based on the edge uv, or G is a 2-dominated chromatic graph i.e. G is a bi-star. In Step 5, we verify whether there exists a vertex of CAN able to dominate the class of color 3. Let x be a vertex of CAN . In Step 5.1, since x is presumed to dominate the class of color 3, we extend the coloring such that x and every vertex not adjacent to x takes a color other than 3. In Step 5.2, all non-colored vertices are adjacent to x, some of them may be adjacent to vertices colored with color 3. We extend the coloring for these last ones. In Step 5.3, the remaining non-colored vertices are adjacent to x and not adjacent to vertices colored with color 3. Thus, an arbitrary coloring does not affect the optimality of final coloring. In Step 5.4, we verify whether the obtained coloring is a dominated coloring, otherwise we continue with another vertex of CAN . Finally, if Step 6 is reached, then G does not admit an edge-3-dominated coloring based on the edge uv.
Algorithm 2 Procedure A Require: A connected graph G = (V, E), an edge e = uv of G. Ensure: The procedure determines whether G admits a 2-dominated coloring or an edge3-dominated coloring based on e. If yes, it gives a minimum dominated coloring of G. Throughout the algorithm, let Nu and Nv be the current sets of non-colored vertices adjacent to u and v, respectively. STEP 1 Give the colors 1 and 2 to u and v, respectively. Let COM = N (u) ∩ N (v), give the color 3 to all vertices of COM . STEP 2 Give the color 2 (resp. the color 1) to the isolated vertices in G[Nu ] (resp. G[Nv ]). Let EXT = V \ (N [u] ∪ N [v]); give the color 3 to all vertices of EXT . STEP 3 for all x ∈ EXT ∪ COM do bicol(x, 3, 2, {x} ∪ Nu ). bicol(x, 3, 1, {x} ∪ Nv ). end for STEP 4 //Forming CAN , the set of candidate vertices to dominate the class of //color 3. CAN = ∅. STEP 4.1 //A dominating vertex of the class of color 3 has to be adjacent to all //vertices already colored with 3 if such vertices exist, otherwise, //to all components of G[Nu ] and G[Nv ]. if EXT ∪ COM 6= ∅ then CAN = {x ∈ V : ∀y ∈ EXT ∪ COM : xy ∈ E} else if h(Nv ) ≥ 2 and h(Nu ) = 1 then CAN = Nu . end if
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if h(Nu ) ≥ 2 and h(Nv ) = 1 then CAN = Nv . end if if h(Nu ) ≤ 1 and h(Nv ) ≤ 1 then CAN = Nu ∪ Nv . end if end if STEP 4.2 if CAN = ∅ then if COM ∪ EXT ∪ Nu ∪ Nv 6= ∅ then Return FALSE. //G does not admit an edge-3-dominated coloring //based on e = uv. else Return TRUE. //G is a 2-dominated chromatic graph. end if end if STEP 5 //Search for a vertex x from CAN to dominate the class of color 3. for all x ∈ CAN do STEP 5.1 //A unique possible extension of the coloring such that x and every //non-adjacent vertex to x receives a color other than 3. for all y ∈ ({x} ∪ (Nu ∪ Nv ) \ N (x)) do if y ∈ Nu then bicol(y, 2, 3, Nu ). end if if y ∈ Nv then bicol(y, 1, 3, Nv ). end if end for STEP 5.2 //A unique possible extension of the coloring such that every non-colored //vertex is adjacent to x and not adjacent to vertices colored with color 3. Let Ax be the component of G[N (u)], or of G[N (v)], containing x. Let A3x be the subset of vertices of Ax colored with 3. for all y ∈ (Nu ∪ Nv ) ∩ N (A3x ) do if y ∈ Nu then bicol(y, 2, 3, Nu ). end if if y ∈ Nv then bicol(y, 1, 3, Nv ). end if end for STEP 5.3 //All non-colored vertices are adjacent to x and non-adjacent to all //vertices colored with 3. We give them an arbitrary proper color. for all y ∈ Nu ∪ Nv do if y ∈ Nu then bicol(y, 2, 3, Nu ). end if if y ∈ Nv then bicol(y, 1, 3, Nv ). end if end for STEP 5.4 //At this step all vertices are colored. if the obtained coloring is proper and x dominate the class of color 3 then Return TRUE. //G admits an edge-3-dominated coloring. end if end for STEP 6 Return FALSE.//G does not admit neither a 2-dominated coloring nor an //edge-3-dominated coloring based on e = uv.
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In the following, we give the description of procedure B.
Procedure B: The procedure B consists of two steps and checks whether G admits a triangle-3-dominated coloring based on an induced triangle K = {x1 , x2 , x3 } of G. For each permutation (i, j, k) of the set {1, 2, 3}, we denote by Pi , the private neighborhood of xi in G with respect to K, i.e. Pi = N (xi ) \ (N (xj ) ∪ N (xk )); and by CNi,j , the common neighborhood of xi and xj except xk , i.e. CNi,j = (N (xi ) ∩ N (xj )) \ {xk }. In the first step, we construct the unique possible coloring. In Step 2, the non-colored vertices belong to the private neighborhood sets. Then, we check whether the obtained coloring is a triangle-3-dominated coloring, clockwise if possible or anti-clockwise, otherwise.
Algorithm 3 Procedure B Require: A connected graph G = (V, E), an induced triangle K = {x1 , x2 , x3 } of G. Ensure: The procedure determines whether G admits a triangle-3-dominated coloring based on K. If yes, it gives a minimum dominated coloring of G. STEP 1 Give the colors 1, 2 and 3 to the vertices x1 , x2 and x3 respectively. for all permutations (i, j, k) of {1, 2, 3} do Give the color k to the vertices of CNi,j . end for STEP 2 Give the color i + 1 (mod 3) to the vertices of Pi , with i : 1..3. if the coloring is proper then Return TRUE. else Give the color i − 1 (mod 3) to the vertices of Pi , with i : 1..3. if the coloring is proper then Return TRUE. end if end if Return FALSE.
Complexity analysis: Let ∆ be the maximum degree and m be the number of edges of G. In the procedure A, the process bicol applied on a component of G[Nu ] or G[Nv ] is in O(∆). Testing whether a coloring is proper and if x dominates the color 3 are done in O(∆2 ) and O(∆), respectively. In the procedure B, O(∆2 ) times are spent for computing the private and the common neighborhoods. The procedure A will be executed O(m) times (corresponding to the complexity of browsing all edges of the graph). The procedure B will be executed O(m · ∆2 ) times (corresponding to the complexity of finding all triangles of the graph). Hence, we need at most O(m ·
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∆2 ) and O(m · ∆4 ) for the first loop and the second loop of Algorithm for 3dominated coloring problem (Algorithm 1), respectively. Hence, the algorithm 3-dominated coloring recognition is polynomial.
Acknowledgments We thank the referees for comments that helped us to improve the paper.
References 1. K. Appel, W. Haken, Every planar map is four colorable, Bulletin of the American Mathematical Society, 82(5), pp. 711-712, 1976. 2. S. Arumugam, K. Raja Chandrasekar, N. Misra, P. Geevarghese, S. Saurabh, Algorithmic aspects of dominator colorings in graphs, IWOCA’2011, LNCS 7056, pp. 19-30, 2011. 3. H. Boumediene Merouane and M. Chellali, On the dominator colorings in trees, Discussiones Mathematicae Graph Theory, 32(4), pp. 677-683, 2012. 4. M. Chellali, F. Maffray, Dominator colorings in some classes of graphs, Graphs and Combinatorics, 28, pp. 97-107, 2012. 5. M. Chellali and L. Volkmann, Relations between the lower domination parameters and the chromatic number of a graph, Discrete Mathematics, 274(1-3), pp.1-8, 2004. 6. G. Chen, A. Gyarfas and R. Schelp, Vertex colorings with a distance restriction, Discrete Mathematics, 191(1-3), pp. 65-82, 1998. 7. R. Gera, On the dominator colorings in bipartite graphs, Information Technology: New Generations, pp. 947-952, 2007. 8. R. Gera, On dominator colorings in graphs, Graph Theory Notes of New York LII, pp. 25-30, 2007. 9. R. Gera, S. Horton and C. Rasmussen, Dominator colorings and safe clique partitions, Congressus Numerantium, 181, pp. 19-32, 2006. 10. M. Haddad and H. Kheddouci. A strict strong coloring of trees, Information Processing Letters, 109(18), pp. 1047-1054, 2009. 11. T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998. 12. T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics, Marcel Dekker, New York, 1998. 13. R. Laskar, J.Pfaff , S. M. Hedetniemi and S. T. Hedetniemi, On the algorithmic complexity of total domination, SIAM Journal on Algebraic and Discrete Methods, 5(3), pp. 420-425, 1984. 14. E. Malaguti and P. Toth. A survey on vertex coloring problems, International Transactions in Operational Research, 17: 1-34, 2010. 15. J. Pfaff, R.C. Laskar, S.T. Hedetniemi, NP-completeness of total and connected domination and irredundance for bipartite graphs, Technical Report 428, Clemson University, Dept. Math. Sciences, 1983.
