On the Infinitesimal Rigidity of Bar-and-Slider Frameworks*
On the Infinitesimal Rigidity of Bar-and-Slider Frameworks Naoki Katoh and Shin-ichi Tanigawa Department of Architecture and Architectural Engineering, Kyoto University, Kyoto Daigaku Katsura, Nishikyo-ku, Kyoto 615-8540 Japan {naoki,is.tanigawa}@archi.kyoto-u.ac.jp
Abstract. A bar-slider framework is a bar-joint framework a part of whose joints are constrained by using line-sliders. Such joints are allowed to move only along the sliders. Streinu and Theran proposed a combinatorial characterization of the infinitesimal rigidity of generic bar-slider frameworks in two dimensional space. In this paper we propose a generalization of their result. In particular, we prove that, even though the directions of the sliders are predetermined and degenerate, i.e., some sliders have the same direction, it is combinatorially decidable whether the framework is infinitesimally rigid or not. Also, in order to prove that, we present a new forest-partition theorem.
1
Introduction
A 2-dimensional bar-joint framework is defined as a pair (G, p), where G = (V, E) is a finite undirected graph having neither loops nor multiple edges and p is a mapping from V to R2 , called a joint configuration. In a framework, each vertex and each edge are regarded as a universal joint and a rigid bar, respectively, and each joint is allowed to move continuously keeping the lengths of the bars. A framework is called flexible if it can be deformed by a continuous motion of joints, and otherwise rigid. A rigid framework is called minimally rigid if removing any bar results in a flexible framework. A common strategy of dealing with the edge length constraints is to take the first order approximation, and this paper also focuses on this rigidity model, called the infinitesimal rigidity. The formal definition will be given in the next section. A joint configuration p is called generic if there is no special algebraic dependency between coordinates of p. (This will be formally defined later.) The celebrated Maxwell-Laman theorem [4] states that, if p is generic, (G, p) is infinitesimally minimally rigid if and only if G satisfies |E| = 2|V | − 3 and |F | ≤ 2|V (F )| − 3 for all nonempty F ⊆ E, where V (F ) denotes the set of vertices spanned by F . A graph satisfying Laman’s counting condition is called a minimally rigid graph or a Laman graph. Instead of Laman’s counting condition,
The first author is supported by Grant-in-Aid for Scientific Research (B) and Grantin-Aid for Scientific Research (C), JSPS. The second author is supported by Grantin-Aid for JSPS Research Fellowships for Young Scientists.
Y. Dong, D.-Z. Du, and O. Ibarra (Eds.): ISAAC 2009, LNCS 5878, pp. 524–533, 2009. c Springer-Verlag Berlin Heidelberg 2009
On the Infinitesimal Rigidity of Bar-and-Slider Frameworks
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(b)
(c)
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(e)
Fig. 1. The sets of bold and dotted edges represent a bipartition. (a) A proper 2forestpartition. (b) A non-proper 2forest-partition. (c) A bar-slider framework. (d) The underlying graph of (c). (e) A proper bipartition satisfying (P1) and (P2) of Proposition 1.
several equivalent characterizations are known, see e.g. [14]. Crapo [1] showed that G is minimally rigid if and only if |E| = 2|V | − 3 holds and E can be partitioned into two colored classes {R, B} such that (i) each color forms a forest, and (ii) no subset V ⊆ V with |V | ≥ 2 induces two colored subtrees that span V simultaneously. A bipartition of E into two forests is called 2forest partition, while a bipartition satisfying the property (ii) is called proper. Hence, a bipartition satisfying both (i) and (ii) is said to be a proper 2forest partition1 (see Figure 1(a)(b)). Streinu and Theran [12] have extended Crapo’s characterization to bar-slider frameworks in a natural way. A bar-slider framework is a bar-joint framework a part of whose joints are constrained by using sliders. As in [12,6], we shall handle each slider as a loop of a graph to extract the combinatorial aspect of frameworks. Let G = (V, E) be an undirected graph that may have some loops, and let us denote the set of loops in F ⊆ E by L(F ) and the set of loops incident to a vertex u ∈ V by δL(E) (u). Then, a bar-slider framework is defined as a triple (G, p, d), where p : V → R2 is a joint configuration and d : L(E) → R2 represents a direction of each slider. Namely, for u ∈ V and e ∈ δL(E) (u), {p(u) + td(e) : t ∈ R} is a line representing a slider incident to p(u), and p(u) is allowed to move along this line (see Figures 1(c) and (d)). A framework is minimally rigid if removing any bar or slider results in a framework that is not rigid. The following proposition is a result of [12]. Proposition 1. ([12]) Let G = (V, E) be an undirected graph. If E can be partitioned into two colored classes {R, B} such that (i) {R \ L(R), B \ L(B)} is a proper 2forest partition2 of E \ L(E), and (ii) each connected component of the graph (V, R) and (V, B) contains exactly one loop of its color, then there exist a joint configuration p and a direction mapping d such that (G, p, d) is infinitesimally rigid. In particular, each of L(R) and L(B) is realized as a slider parallel to the x-axis and the y-axis, respectively. Figure 1(e) shows an example of a partition of E satisfying (P1) and (P2). We refer to it as a proper 2rooted-forest partition. As a corollary of the algorithm by 1
2
A partition of E into three trees such that each vertex is incident to exactly two of them is called a 3tree2 partition, and Crapo’s partition is usually called a proper 3tree2 partition. It is known that E admits a proper 3tree2 partition if and only if E admits a proper 2forest partition with |E| = 2|V | − 3. In [12], this is called an induced-cut 2-forest in order to emphasize the existence of a monochromatic cut in any induced subgraph.
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Streinu and Theran [13] for checking the sparsity of a graph, it is known that E admits a proper 2rooted-forest partition if and only if (L1) |E| = 2|V |, (L2) |F | ≤ 2|V (F )| − 3 for every nonempty F ⊆ E \ L(E), and (L3) |F | ≤ 2|V (F )| for every F ⊆ E. In this paper, we will provide an extension of these results. Proposition 1 says that, if E admits a proper 2rooted-forest partition {R, B}, then each of L(R) and L(B) is realized as an x-slider and a y-slider, respectively. It is not however obvious whether a specified loop is realized as either x-or y-slider until we actually construct a partition specifically. In most practical situations, the directions of sliders are predetermined, and then we are not allowed to realize the predetermined x-slider as a y-slider or vice verse. This raises the following question. Given a set of joints connected by some bars as well as some sliders whose directions are specified and moreover some of which may have the same direction (e.g. x-direction or y-direction), we would like to decide whether it is rigid or flexible. Notice that, even though a joint configuration is generic, (G, p, d) could be either rigid or flexible depending on d. In this paper we will prove, however, that the generic rigidity does not actually depend on the specific values of d, and it is combinatorially decidable whether a framework is rigid or not even though the directions of the sliders are “predetermined” and “degenerate”. To extract a combinatorial aspect of this problem, we shall consider a loopcolored graph Gc = (V, E, c), where c is a mapping from L(E) to a finite set C of colors. Each color indicates a direction of a slider. We then redefine a bar-slider framework as a triple (Gc , p, d), where Gc is a loop-colored graph, p is a joint configuration, and d is a direction mapping from C to R2 (not from L(E)) such that d(c) and d(c ) are linearly independent for any pair of distinct colors c and c in C. A loop colored in c ∈ C is supposed to be realized as a slider with the direction d(c) (see Figure 2). A main result of this paper is stated as follows. Theorem 1. Let Gc = (V, E, c) be a loop-colored graph. Then, for any direction mapping d and for any generic joint-configuration p, the bar-slider framework (Gc , d, p) is infinitesimally rigid if and only if Gc satisfies (L1)∼(L3) as well as (L4) |F | ≤ 2|V (F )|−1 for any F ⊆ E such that all loops of L(F ) are monochromatically colored. The necessity of Theorem 1 follows straightforwardly from the definition of the rigidity. In Section 3, we shall propose nontrivial forest-partition theorems (Theorem 2 and Theorem 3), which might be interesting results in their own right. In Section 4, we will provide a proof of the sufficiency of Theorem 1 based on the forest-partition theorem. As for the related works of bar-slider frameworks, we should mention the pinning problem of a bar-joint framework in the plane. In this problem, given a bar-joint framework (having certain degree of freedom), we would like to stabilize it by fixing the positions of the smallest number of joints. Lov´ asz [7] showed, as
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Fig. 2. A loop-colored graph Gc and a bar-slider framework (Gc , p, d)
an application of his matroid matching algorithm, that the pinning problem can be reduced to a 2-polymatroid matching problem, and is solvable in polynomial time. Fekete [2] provided a simpler min-max characterization of the optimal value in generic case. Fekete and Jord´an further discussed in [3] the pinning problem from the view point of generic global rigidity. Also, Servatius, Shai and Whiteley [10] have presented a counting condition for the pinned bar-joint framework. Notice that pinning down a joint reduces the degree of freedom of a framework by at most two, while attaching a slider at some joint reduces the degree of freedom by at most one. In fact, attaching a slider seems easier to handle than pinning a joint and we thereby obtain a much clearer and extended combinatorial characterization of the rigidity of bar-slider frameworks.
2
Preliminaries
Matroids and Submodular Functions. We skip the definition, basic terminologies and fundamental properties of matroids (see e.g. [8]). Let E be a finite set. The function f : 2E → R is called submodular if f (X) + f (Y ) ≥ f (X ∪ Y ) + f (X ∩ Y ) for any X, Y ∈ 2E and nondecreasing if f (X) ≤ f (Y ) for any X ⊆ Y ⊆ E. Let f : 2E → Z be an integer-valued nondecreasing submodular function. It is known that f induces a matroid on E, denoted by Mf , whose collection of independent sets is written by I(Mf ) = {I ⊆ E : |I | ≤ f (I ) for nonempty I ⊆ I} see e.g., [8, Chapter 12]. For an edge set F , P(F ) denotes the collection of all possible partitions {F0 , F1 , . . . , Fm } of F for some integer m with 0 ≤ m ≤ |F | such that Fi
= ∅ for each i = 1, . . . , m (and F0 may be empty). The following proposition provides an explicit formula expressing the rank function rf of Mf , which is in the form of the Dilworth truncation (restricted to a matroid), see e.g. [11, Chapter 48]. Proposition 2. Let f be an integer-valued nondecreasing submodular function on E satisfying f (F ) ≥ 0 for every nonempty F ⊆ E. Then, for any nonempty F ⊆ E, the rank rf (F ) of F in Mf is given by rf (F ) =
min
{|F0 | +
{F0 ,...,Fm }∈P(F )
m
f (Fi )}.
(1)
i=1
Let us consider the matroid union Mf ∨ Mg of Mf and Mg induced by integervalued nondecreasing submodular functions f and g on E. Pym and Perfect [9] showed that Mf ∨ Mg is the matroid induced by the submodular function
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f + g, i.e. Mf ∨ Mg = Mf +g , if f (F ) ≥ 0 and g(F ) ≥ 0 hold for every F ⊆ E including ∅. Whiteley further claimed Mf ∨ Mg = Mf +g in [14] even in the case of f (∅) < 0 or g(∅) < 0. Although this statement is true for the union of the same matroids, say the union of graphic matroids, Jord´ an pointed out that this is not always true in general. We shall show below a sufficient condition for the statement to be true. Lemma 1. Let f and g be integer-valued nondecreasing submodular functions on E satisfying f (F ) ≥ 0 and g(F ) ≥ 0 for every nonempty F ⊆ E. Then, Mf ∨ Mg = Mf +g holds if, for any F ⊆ E, there exists a partition {F0 , F1 , . . . , Fm } ∈ P(F ) that takes the minimum values of (1) for rf (F ) and rg (F ) simultaneously. Bar-joint Rigidity. Recall that a bar-joint framework is defined as a pair (G, p) of a graph G and a joint configuration p : V → R2 . An infinitesimal motion of (G, p) is defined as an assignment v : V → R2 of a 2-dimensional vector for each joint p(v) such that (p(v) − p(u)) · (v(v) − v(u)) = 0
for each uv ∈ E.
(2)
We refer to (2) as the length constraint by the bar p(u)p(v). Collecting (2) for all e ∈ E, we have a system of the |E| equations on the unknown v(u), u ∈ V . Hence v is an infinitesimal motion if and only if it is in the null space of the |E| × 2|V |-matrix R(G, p), so-called the rigidity matrix of (G, p). If |V | ≥ 2 and the rank of R(G, p) is equal to 2|V | − 3, the framework (G, p) is said to be infinitesimally rigid. Equivalently, (G, p) is infinitesimally rigid if and only if all solutions of R(G, p)v = 0 are trivial, that is, (derivatives of) translations and rotations of the whole framework, see e.g. [14] for more details. A configuration p is called generic with respect to G if the rank of the rigidity matrix R(G, p) and those of all its row-induced submatrices have the maximum values taken over all configurations p. Note that each minor of the rigidity matrix is written as a polynomial of coordinates of p. If such a polynomial is not identically zero, then it takes nonzero value for almost all joint configurations. Namely, a set of generic joint configurations forms an open dense subset of the space of joint configurations. (see e.g. [14]). Therefore, in almost all cases, the rigidity of frameworks is completely determined by the underlying graphs. Bar-slider Rigidity. Recall that a bar-slider framework is defined as a triple (Gc , p, d), where Gc is a loop-colored graph, and d is a direction mapping from a set of colors to R2 . A joint p(u) is constrained to be on the line {p(u)+td(c(e)) : t ∈ R} for each e ∈ δL(E) (u). Rigidity of a bar-slider framework is defined in a similar way as in the case of a bar-joint framework, but it counts even trivial motions as its degree of freedom. (Gc , p, d) is rigid if there exists no continuous motion of p which converts to a distinct framework under bar length constraints as well as slider constraints. Again, we shall consider the first-order rigidity of this concept, where an
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infinitesimal motion v : V → R2 of (Gc , p, d) satisfies (2) as well as direction constraints written as v(u) · d(c(e))⊥ = 0
for each u ∈ V and e ∈ δL(E) (u),
(3)
⊥
where d(c(e)) denotes a vector orthogonal to d(c(e)). As a result, taking new rows corresponding to the direction constraints (3) into account, we obtain the rigidity matrix R(Gc , p, d), whose size becomes |E| × 2|V |. It is called infinitesimally rigid if no infinitesimal motion exists (except for 0), equivalently the rank of R(Gc , p, d) is equal to 2|V |. A generic joint configuration p with respect to a given Gc and a given d is defined as in the case of bar-joint frameworks.
3
Combinatorial Results
Let Gc = (V, E, c) be a loop-colored graph, and {c1 , c2 , . . . , ck } be the set of colors appearing in Gc . By regarding each self-loop as a root, a subgraph G = (V , E ) of Gc is said to be a rooted-forest colored in ci if (i) (V , E \ L(E )) is a forest, (ii) E does not contain any loop colored in cj with j = i, and (iii) each connected component of G contains exactly one loop colored in ci . G is further called a spanning rooted-forest colored in ci if E spans V . We say that Gc satisfies the strong counting condition if it satisfies (L1)∼(L4) given in the introduction. In this section we shall reveal properties of graphs satisfying the strong counting condition. In particular, we generalize a concept of the forest partitions to a partition E = {E1 , E2 , . . . , Ek } of E into k components such that (P1) each Ei induces a rooted-forest colored in ci , (P2) each vertex is spanned by exactly two components, i.e., |{i : δE (v) ∩ Ei = ∅}| = 2 holds for each v ∈ V . We refer to each component of E as a colored class. If a partition of E satisfies these two conditions, we say that E (or Gc ) admits a (k, 2)-rooted-forest partition. As before, a (k, 2)-rooted-forest partition is said to be proper if no two subtrees from Ei \ L(Ei ) and Ej \ L(Ej ) span the same set of vertices for any 1 ≤ i, j ≤ k with i
= j. Figure 3 shows examples of (k, 2)-rooted-forest partitions. The following forest partition theorem is our main combinatorial result. Theorem 2. Let Gc be a loop-colored graph, and let k be the number of colors used in Gc . Then, Gc satisfies the strong counting condition if and only if it admits a proper (k, 2)-rooted-forest partition. To prove Theorem 2, we define a counting condition weaker than (L2) as (L2’) |F | ≤ 2|V (F )| − 2 for every nonempty F ⊆ E \ L(E), and let us refer to the set of the counting conditions (L1), (L2’), (L3) and (L4) as the weak counting condition. Although the detailed description is omitted, Theorem 2 easily follows from the next result.
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(a)
(b)
Fig. 3. (a) A proper (k, 2)-rooted-forest partition for k = 3. (b) A non-proper (k, 2)rooted-forest partition for k = 3.
Theorem 3. Let Gc be a loop-colored graph, and let k be the number of colors used in Gc . Then, Gc satisfies the weak counting condition if and only if it admits a (k, 2)-rooted-forest partition. Notice that, a set of edges satisfying the weak counting condition and the strong counting condition, respectively, is a base of the matroid induced by the integervalued nondecreasing submodular functions μ : 2E → Z and μ : 2E → Z, respectively, defined as μ(F ) = 2|V (F )| − 2 + min{χ(F ), 2}
μ (F ) =
2|V (F )| − 3 2|V (F )| − 2 + min{χ(F ), 2}
if L(F ) = ∅ otherwise,
(4)
(5)
where χ(F ) denotes the total number of colors appearing in L(F ). Therefore, Theorem 3 implies that a set of edges admitting a (k, 2)-rooted-forest partition is characterized in terms of a matroid as the well-known characterization of forestpartitions in terms of the union of graphic matroids. Also, Theorem 2 generalizes Crapo’s characterization [1] of Laman graphs. Proof of Theorem 3 for k = 2. We now consider a special case of Theorem 3 where the number of colors k is restricted to two. We prove that (2, 2)-rootedforest partitions can be characterized in terms of the union of two matroids. For a vertex set V , let K(V ) denote the complete graph on V , and let K + (V ) denotes the graph obtained from K(V ) by attaching two loops at each vertex. We simply denote by K + (V ) the edge set of K + (V ), if it is clear from the context. Throughout this restricted case (of k = 2), we assume that one of two loops incident to v in K + (V ) is colored in red and the other one is colored in blue for each v ∈ V . For an edge set F ⊆ K + (V ), let Lr (F ) and Lb (F ) denote the sets of loops colored in red and blue, respectively. + Let us first consider the following function τr : 2K (V ) → Z; For F ⊆ K + (V ), τr (F ) = |V (F )| − 1 + χr (F ),
(6)
where χr (F ) is defined as χr (F ) = 0 if Lr (F ) = ∅ and otherwise χr (F ) = 1. Then, it is not difficult to see that τr is submodular. Also τr is nondecreasing,
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and hence it induces a matroid, denoted by Mτr , on K + (V ). The functions τb and χb , and the matroid Mτb (for the blue color) are symmetrically defined. Lemma 2. Let c ∈ {r, b} and c¯ ∈ {r, b} \ {c}. An edge set F of K + (V ) is a base of Mτc if and only if it is a spanning rooted-forest colored in c. Let us consider how the independent sets of Mτr ∨ Mτb can be characterized in terms of the counting condition. To apply Lemma 1, we just need to show the following property. Lemma 3. Let c ∈ {r, b} and let F ⊆ K + (V ). Let m be the total number of connected components of G[F ] and {F1 , . . . , Fm } be a partition of F such that, for each j = 1, . . . , m, Fj is the edge set of a connected component of G[F ]. Also, let F0 = ∅. Then, {F0 , F1 , . . . , Fm } ∈ P(F ) takes theminimum value of m (1). Namely, the rank rτc (F ) can be described by rτc (F ) = i=1 τc (Fi ). Combining Lemma 1 and Lemma 3, we obtain Mτr ∨ Mτb = Mτr +τb , and the following lemma easily follows. Lemma 4. Let E ⊆ K + (V ). Then, E is a base of Mτr ∨ Mτb if and only if it satisfies the weak counting condition. We are now ready to show Theorem 3 for k = 2. By Lemma 2, an edge set is a base of Mτr ∨ Mτb if and only if it can be partitioned into Er and Eb which are spanning rooted-forests colored in red and blue, respectively, and equivalently it admits a (2, 2)-rooted-forest partition. Combining Lemma 4 with this fact, we conclude that an edge set admits a (2, 2)-rooted-forest partition if and only if it satisfies the weak counting condition. Due to the space limitation, we omit the proof of Theorem 3 for k > 2 in this extended abstract. Other Combinatorial Results. In order to prove Theorem 1 we need one more combinatorial lemma related to the weak counting condition. We refer to a vertex v as a (a, b)-vertex in Gc if δE\L(E) (v) = a and δL(E) (v) = b. The following lemma claims the existence of small degree vertices. Lemma 5. Let Gc = (V, E, c) be a connected graph with |V | > 1 satisfying the weak counting condition. Suppose that there exists no (1, 1), (1, 2), (2, 0), (3, 0)vertex. Then, for any (k, 2)-rooted-forest partition E of Gc , there exists a (2, 1)vertex v such that the two non-loop edges incident to v belong to distinct colored classes in E. Remark. Each condition of the strong counting condition (and also the weak one) can be checked by using the so-called “pebble game” algorithm for checking the sparsity of graphs (see [5,13]). Since the pebble game works in O(|V |2 ) time, checking whether Gc satisfies the strong counting condition (also the weak one) can be done in O(k|V |2 ) time, where k is the number of colors. This can be improved to O(|V |2 ) time by just avoiding to start each pebble game from scratch. Therefore, assuming a generic joint configuration, we can decide in O(|V |2 ) time whether a bar-slider framework is rigid or not.
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Infinitesimally Rigid Bar-Slider Frameworks
We now prove the following statement; for a given loop-colored graph Gc satisfying the strong counting condition and a direction mapping d on a set of colors, there exists a joint configuration p such that the bar-slider framework (Gc , p, d) is infinitesimally rigid in the plane. Note that, if one particular realization (Gc , p, d) is rigid, then (Gc , q, d) becomes rigid for all generic joint configurations q as explained in Section 2, implying the nontrivial part (sufficiency) of Theorem 1. The proof is done by induction on |E \ L(E)| as follows. We convert Gc to a slightly smaller graph Gc with respect to |E \ L(E)|, and then prepare a rigid realization (Gc , p, d) based on the induction hypothesis. Finally, we shall show that (Gc , p, d) is infinitesimally rigid. We omit the base case. Let us consider the case of |E \ L(E)| > 0. Applying Theorem 2, we have a proper (k, 2)-rooted-forest partition E = {E1 , E2 , . . . , Ek } of Gc , where k = χ(E). For the convenience of the description, let us assume that all edges of Gc (not only loops but also non-loop edges) are colored according to this partition E throughout the proof. Following Lemma 5, the proof is split into five cases depending on the existence of small degree vertices. Due to the space limitation, we shall omit the four cases, and show only the final case where Gc has no (1, 1), (1, 2), (2, 0), (3, 0)-vertex. In this case, by Lemma 5, Gc has a (2, 1)-vertex v such that the two non-loop edges incident to v belong to distinct colored classes in E. Let e1 be the loop attached to v with the color c1 , and let va and vb be the two non-loop edges incident to v. By condition (P2) of E and Lemma 5, we may assume that va and vb are colored in c1 and c2 , respectively (see Figure 4). We shall consider the graph Gc obtained from Gc by removing va, vb and then attaching new loops f1 colored in c1 to a and f2 colored in c2 to v, respectively (see Figure 4). Then, the coloring of the edge set induces a proper (k, 2)-rootedforest partition of Gc , implying that Gc satisfies the strong counting condition. By induction, there exists a rigid realization (Gc , p, d). Since the joint p(v) is isolated, we may assume that p(v) is located such that p(v) − p(a) is orthogonal to the direction d(c1 ) (as shown in Figure 4). We claim that (Gc , p, d) is infinitesimally rigid. To see this, suppose that there exists a nonzero infinitesimal motion v for (Gc , p, d). Due to the direction constraint by the slider associated with e1 , v(v) is a scalar multiple of d(c1 ), and consequently v(a) is also a scalar multiple of d(c1 ) by the length constraint of the bar p(v)p(a). Let us define v : V → R2 as v (v) = 0 and v (u) = v(u) for
e1
e1
v b
a
f1
p(v)
v f2
a
Gc
G’c’
realization
b
d(c1)
p(v)
d(c1) p(a)
p(b)
(G’c’ ,p,d)
Fig. 4. (2, 1)-vertex
p(b) p(a)
(Gc ,p,d)
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all u ∈ V \ {v}. Then, it is not difficult to see that v satisfies all the constraints appearing in (Gc , p, d) since v satisfies the direction constraints by the sliders associated with f1 and f2 , only which are not contained in Gc . Therefore, all entries of v must be zero because (Gc , p, d) is rigid. Since v is nonzero, we obtain that v(v)
= 0 and v(u) = v (u) = 0 for all u ∈ V \ {v}. However, v does not satisfy the length constraint (2) by the bar p(v)p(b) which (Gc , p, d) has. This contradicts that v is an infinitesimal motion of (Gc , p, d).
References 1. Crapo, H.: On the generic rigidity of plane frameworks. Technical Report 1278, Institute de recherche d’informatique et d’automatique (1988) 2. Fekete, Z.: Source location with rigidity and tree packing requirements. Operations Research Letters 34(6), 607–612 (2006) 3. Fekete, Z., Jord´ an, T.: Uniquely localizable networks with few anchors. In: Nikoletseas, S.E., Rolim, J.D.P. (eds.) ALGOSENSORS 2006. LNCS, vol. 4240, pp. 176–183. Springer, Heidelberg (2006) 4. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4, 331–340 (1970) 5. Lee, A., Streinu, I.: Pebble game algorithms and sparse graphs. Discrete Mathematics 308(8), 1425–1437 (2008) 6. Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proc. 19th Canadian Conference on Computational Geometry, CCCG 2007 (2007) 7. Lov´ asz, L.: Matroid matching and some applications. J. Combinatorial Theory, Series (B) 28, 208–236 (1980) 8. Oxley, J.: Matroid Theory. Oxford University Press, Oxford (1992) 9. Pym, J.S., Perfect, H.: Submodular functions and independence structures. J. Math. Analysis Appl. 30, 1–31 (1970) 10. Servatius, B., Shai, O., Whiteley, W.: Combinatorial characterization of the assur graphs from engineering. arXiv:0801.252v1 (2008) 11. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003) 12. Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: Proc. ACM Symposium on Computational Geometry (SoCG 2008), pp. 365–374 (2008) 13. Streinu, I., Theran, L.: Sparsity-certifying graph decompositions. Graphs and Combinatorics 25(2), 219–238 (2009) 14. Whiteley, W.: Matroids from discrete geometry. In: Bonin, J., Oxley, J., Servatius, B. (eds.) Matroid Theory. AMS Contemporary Mathematics, pp. 171–313 (1997)
Abstract. A bar-slider framework is a bar-joint framework a part of whose joints are constrained by using line-sliders. Such joints are allowed to move only along the sliders. Streinu and Theran proposed a combinatorial characterization of the infinitesimal rigidity of generic bar-slider frameworks in two dimensional space. In this paper we propose a generalization of their result. In particular, we prove that, even though the directions of the sliders are predetermined and degenerate, i.e., some sliders have the same direction, it is combinatorially decidable whether the framework is infinitesimally rigid or not. Also, in order to prove that, we present a new forest-partition theorem.
1
Introduction
A 2-dimensional bar-joint framework is defined as a pair (G, p), where G = (V, E) is a finite undirected graph having neither loops nor multiple edges and p is a mapping from V to R2 , called a joint configuration. In a framework, each vertex and each edge are regarded as a universal joint and a rigid bar, respectively, and each joint is allowed to move continuously keeping the lengths of the bars. A framework is called flexible if it can be deformed by a continuous motion of joints, and otherwise rigid. A rigid framework is called minimally rigid if removing any bar results in a flexible framework. A common strategy of dealing with the edge length constraints is to take the first order approximation, and this paper also focuses on this rigidity model, called the infinitesimal rigidity. The formal definition will be given in the next section. A joint configuration p is called generic if there is no special algebraic dependency between coordinates of p. (This will be formally defined later.) The celebrated Maxwell-Laman theorem [4] states that, if p is generic, (G, p) is infinitesimally minimally rigid if and only if G satisfies |E| = 2|V | − 3 and |F | ≤ 2|V (F )| − 3 for all nonempty F ⊆ E, where V (F ) denotes the set of vertices spanned by F . A graph satisfying Laman’s counting condition is called a minimally rigid graph or a Laman graph. Instead of Laman’s counting condition,
The first author is supported by Grant-in-Aid for Scientific Research (B) and Grantin-Aid for Scientific Research (C), JSPS. The second author is supported by Grantin-Aid for JSPS Research Fellowships for Young Scientists.
Y. Dong, D.-Z. Du, and O. Ibarra (Eds.): ISAAC 2009, LNCS 5878, pp. 524–533, 2009. c Springer-Verlag Berlin Heidelberg 2009
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(a)
(b)
(c)
(d)
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(e)
Fig. 1. The sets of bold and dotted edges represent a bipartition. (a) A proper 2forestpartition. (b) A non-proper 2forest-partition. (c) A bar-slider framework. (d) The underlying graph of (c). (e) A proper bipartition satisfying (P1) and (P2) of Proposition 1.
several equivalent characterizations are known, see e.g. [14]. Crapo [1] showed that G is minimally rigid if and only if |E| = 2|V | − 3 holds and E can be partitioned into two colored classes {R, B} such that (i) each color forms a forest, and (ii) no subset V ⊆ V with |V | ≥ 2 induces two colored subtrees that span V simultaneously. A bipartition of E into two forests is called 2forest partition, while a bipartition satisfying the property (ii) is called proper. Hence, a bipartition satisfying both (i) and (ii) is said to be a proper 2forest partition1 (see Figure 1(a)(b)). Streinu and Theran [12] have extended Crapo’s characterization to bar-slider frameworks in a natural way. A bar-slider framework is a bar-joint framework a part of whose joints are constrained by using sliders. As in [12,6], we shall handle each slider as a loop of a graph to extract the combinatorial aspect of frameworks. Let G = (V, E) be an undirected graph that may have some loops, and let us denote the set of loops in F ⊆ E by L(F ) and the set of loops incident to a vertex u ∈ V by δL(E) (u). Then, a bar-slider framework is defined as a triple (G, p, d), where p : V → R2 is a joint configuration and d : L(E) → R2 represents a direction of each slider. Namely, for u ∈ V and e ∈ δL(E) (u), {p(u) + td(e) : t ∈ R} is a line representing a slider incident to p(u), and p(u) is allowed to move along this line (see Figures 1(c) and (d)). A framework is minimally rigid if removing any bar or slider results in a framework that is not rigid. The following proposition is a result of [12]. Proposition 1. ([12]) Let G = (V, E) be an undirected graph. If E can be partitioned into two colored classes {R, B} such that (i) {R \ L(R), B \ L(B)} is a proper 2forest partition2 of E \ L(E), and (ii) each connected component of the graph (V, R) and (V, B) contains exactly one loop of its color, then there exist a joint configuration p and a direction mapping d such that (G, p, d) is infinitesimally rigid. In particular, each of L(R) and L(B) is realized as a slider parallel to the x-axis and the y-axis, respectively. Figure 1(e) shows an example of a partition of E satisfying (P1) and (P2). We refer to it as a proper 2rooted-forest partition. As a corollary of the algorithm by 1
2
A partition of E into three trees such that each vertex is incident to exactly two of them is called a 3tree2 partition, and Crapo’s partition is usually called a proper 3tree2 partition. It is known that E admits a proper 3tree2 partition if and only if E admits a proper 2forest partition with |E| = 2|V | − 3. In [12], this is called an induced-cut 2-forest in order to emphasize the existence of a monochromatic cut in any induced subgraph.
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Streinu and Theran [13] for checking the sparsity of a graph, it is known that E admits a proper 2rooted-forest partition if and only if (L1) |E| = 2|V |, (L2) |F | ≤ 2|V (F )| − 3 for every nonempty F ⊆ E \ L(E), and (L3) |F | ≤ 2|V (F )| for every F ⊆ E. In this paper, we will provide an extension of these results. Proposition 1 says that, if E admits a proper 2rooted-forest partition {R, B}, then each of L(R) and L(B) is realized as an x-slider and a y-slider, respectively. It is not however obvious whether a specified loop is realized as either x-or y-slider until we actually construct a partition specifically. In most practical situations, the directions of sliders are predetermined, and then we are not allowed to realize the predetermined x-slider as a y-slider or vice verse. This raises the following question. Given a set of joints connected by some bars as well as some sliders whose directions are specified and moreover some of which may have the same direction (e.g. x-direction or y-direction), we would like to decide whether it is rigid or flexible. Notice that, even though a joint configuration is generic, (G, p, d) could be either rigid or flexible depending on d. In this paper we will prove, however, that the generic rigidity does not actually depend on the specific values of d, and it is combinatorially decidable whether a framework is rigid or not even though the directions of the sliders are “predetermined” and “degenerate”. To extract a combinatorial aspect of this problem, we shall consider a loopcolored graph Gc = (V, E, c), where c is a mapping from L(E) to a finite set C of colors. Each color indicates a direction of a slider. We then redefine a bar-slider framework as a triple (Gc , p, d), where Gc is a loop-colored graph, p is a joint configuration, and d is a direction mapping from C to R2 (not from L(E)) such that d(c) and d(c ) are linearly independent for any pair of distinct colors c and c in C. A loop colored in c ∈ C is supposed to be realized as a slider with the direction d(c) (see Figure 2). A main result of this paper is stated as follows. Theorem 1. Let Gc = (V, E, c) be a loop-colored graph. Then, for any direction mapping d and for any generic joint-configuration p, the bar-slider framework (Gc , d, p) is infinitesimally rigid if and only if Gc satisfies (L1)∼(L3) as well as (L4) |F | ≤ 2|V (F )|−1 for any F ⊆ E such that all loops of L(F ) are monochromatically colored. The necessity of Theorem 1 follows straightforwardly from the definition of the rigidity. In Section 3, we shall propose nontrivial forest-partition theorems (Theorem 2 and Theorem 3), which might be interesting results in their own right. In Section 4, we will provide a proof of the sufficiency of Theorem 1 based on the forest-partition theorem. As for the related works of bar-slider frameworks, we should mention the pinning problem of a bar-joint framework in the plane. In this problem, given a bar-joint framework (having certain degree of freedom), we would like to stabilize it by fixing the positions of the smallest number of joints. Lov´ asz [7] showed, as
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Fig. 2. A loop-colored graph Gc and a bar-slider framework (Gc , p, d)
an application of his matroid matching algorithm, that the pinning problem can be reduced to a 2-polymatroid matching problem, and is solvable in polynomial time. Fekete [2] provided a simpler min-max characterization of the optimal value in generic case. Fekete and Jord´an further discussed in [3] the pinning problem from the view point of generic global rigidity. Also, Servatius, Shai and Whiteley [10] have presented a counting condition for the pinned bar-joint framework. Notice that pinning down a joint reduces the degree of freedom of a framework by at most two, while attaching a slider at some joint reduces the degree of freedom by at most one. In fact, attaching a slider seems easier to handle than pinning a joint and we thereby obtain a much clearer and extended combinatorial characterization of the rigidity of bar-slider frameworks.
2
Preliminaries
Matroids and Submodular Functions. We skip the definition, basic terminologies and fundamental properties of matroids (see e.g. [8]). Let E be a finite set. The function f : 2E → R is called submodular if f (X) + f (Y ) ≥ f (X ∪ Y ) + f (X ∩ Y ) for any X, Y ∈ 2E and nondecreasing if f (X) ≤ f (Y ) for any X ⊆ Y ⊆ E. Let f : 2E → Z be an integer-valued nondecreasing submodular function. It is known that f induces a matroid on E, denoted by Mf , whose collection of independent sets is written by I(Mf ) = {I ⊆ E : |I | ≤ f (I ) for nonempty I ⊆ I} see e.g., [8, Chapter 12]. For an edge set F , P(F ) denotes the collection of all possible partitions {F0 , F1 , . . . , Fm } of F for some integer m with 0 ≤ m ≤ |F | such that Fi
= ∅ for each i = 1, . . . , m (and F0 may be empty). The following proposition provides an explicit formula expressing the rank function rf of Mf , which is in the form of the Dilworth truncation (restricted to a matroid), see e.g. [11, Chapter 48]. Proposition 2. Let f be an integer-valued nondecreasing submodular function on E satisfying f (F ) ≥ 0 for every nonempty F ⊆ E. Then, for any nonempty F ⊆ E, the rank rf (F ) of F in Mf is given by rf (F ) =
min
{|F0 | +
{F0 ,...,Fm }∈P(F )
m
f (Fi )}.
(1)
i=1
Let us consider the matroid union Mf ∨ Mg of Mf and Mg induced by integervalued nondecreasing submodular functions f and g on E. Pym and Perfect [9] showed that Mf ∨ Mg is the matroid induced by the submodular function
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f + g, i.e. Mf ∨ Mg = Mf +g , if f (F ) ≥ 0 and g(F ) ≥ 0 hold for every F ⊆ E including ∅. Whiteley further claimed Mf ∨ Mg = Mf +g in [14] even in the case of f (∅) < 0 or g(∅) < 0. Although this statement is true for the union of the same matroids, say the union of graphic matroids, Jord´ an pointed out that this is not always true in general. We shall show below a sufficient condition for the statement to be true. Lemma 1. Let f and g be integer-valued nondecreasing submodular functions on E satisfying f (F ) ≥ 0 and g(F ) ≥ 0 for every nonempty F ⊆ E. Then, Mf ∨ Mg = Mf +g holds if, for any F ⊆ E, there exists a partition {F0 , F1 , . . . , Fm } ∈ P(F ) that takes the minimum values of (1) for rf (F ) and rg (F ) simultaneously. Bar-joint Rigidity. Recall that a bar-joint framework is defined as a pair (G, p) of a graph G and a joint configuration p : V → R2 . An infinitesimal motion of (G, p) is defined as an assignment v : V → R2 of a 2-dimensional vector for each joint p(v) such that (p(v) − p(u)) · (v(v) − v(u)) = 0
for each uv ∈ E.
(2)
We refer to (2) as the length constraint by the bar p(u)p(v). Collecting (2) for all e ∈ E, we have a system of the |E| equations on the unknown v(u), u ∈ V . Hence v is an infinitesimal motion if and only if it is in the null space of the |E| × 2|V |-matrix R(G, p), so-called the rigidity matrix of (G, p). If |V | ≥ 2 and the rank of R(G, p) is equal to 2|V | − 3, the framework (G, p) is said to be infinitesimally rigid. Equivalently, (G, p) is infinitesimally rigid if and only if all solutions of R(G, p)v = 0 are trivial, that is, (derivatives of) translations and rotations of the whole framework, see e.g. [14] for more details. A configuration p is called generic with respect to G if the rank of the rigidity matrix R(G, p) and those of all its row-induced submatrices have the maximum values taken over all configurations p. Note that each minor of the rigidity matrix is written as a polynomial of coordinates of p. If such a polynomial is not identically zero, then it takes nonzero value for almost all joint configurations. Namely, a set of generic joint configurations forms an open dense subset of the space of joint configurations. (see e.g. [14]). Therefore, in almost all cases, the rigidity of frameworks is completely determined by the underlying graphs. Bar-slider Rigidity. Recall that a bar-slider framework is defined as a triple (Gc , p, d), where Gc is a loop-colored graph, and d is a direction mapping from a set of colors to R2 . A joint p(u) is constrained to be on the line {p(u)+td(c(e)) : t ∈ R} for each e ∈ δL(E) (u). Rigidity of a bar-slider framework is defined in a similar way as in the case of a bar-joint framework, but it counts even trivial motions as its degree of freedom. (Gc , p, d) is rigid if there exists no continuous motion of p which converts to a distinct framework under bar length constraints as well as slider constraints. Again, we shall consider the first-order rigidity of this concept, where an
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infinitesimal motion v : V → R2 of (Gc , p, d) satisfies (2) as well as direction constraints written as v(u) · d(c(e))⊥ = 0
for each u ∈ V and e ∈ δL(E) (u),
(3)
⊥
where d(c(e)) denotes a vector orthogonal to d(c(e)). As a result, taking new rows corresponding to the direction constraints (3) into account, we obtain the rigidity matrix R(Gc , p, d), whose size becomes |E| × 2|V |. It is called infinitesimally rigid if no infinitesimal motion exists (except for 0), equivalently the rank of R(Gc , p, d) is equal to 2|V |. A generic joint configuration p with respect to a given Gc and a given d is defined as in the case of bar-joint frameworks.
3
Combinatorial Results
Let Gc = (V, E, c) be a loop-colored graph, and {c1 , c2 , . . . , ck } be the set of colors appearing in Gc . By regarding each self-loop as a root, a subgraph G = (V , E ) of Gc is said to be a rooted-forest colored in ci if (i) (V , E \ L(E )) is a forest, (ii) E does not contain any loop colored in cj with j = i, and (iii) each connected component of G contains exactly one loop colored in ci . G is further called a spanning rooted-forest colored in ci if E spans V . We say that Gc satisfies the strong counting condition if it satisfies (L1)∼(L4) given in the introduction. In this section we shall reveal properties of graphs satisfying the strong counting condition. In particular, we generalize a concept of the forest partitions to a partition E = {E1 , E2 , . . . , Ek } of E into k components such that (P1) each Ei induces a rooted-forest colored in ci , (P2) each vertex is spanned by exactly two components, i.e., |{i : δE (v) ∩ Ei = ∅}| = 2 holds for each v ∈ V . We refer to each component of E as a colored class. If a partition of E satisfies these two conditions, we say that E (or Gc ) admits a (k, 2)-rooted-forest partition. As before, a (k, 2)-rooted-forest partition is said to be proper if no two subtrees from Ei \ L(Ei ) and Ej \ L(Ej ) span the same set of vertices for any 1 ≤ i, j ≤ k with i
= j. Figure 3 shows examples of (k, 2)-rooted-forest partitions. The following forest partition theorem is our main combinatorial result. Theorem 2. Let Gc be a loop-colored graph, and let k be the number of colors used in Gc . Then, Gc satisfies the strong counting condition if and only if it admits a proper (k, 2)-rooted-forest partition. To prove Theorem 2, we define a counting condition weaker than (L2) as (L2’) |F | ≤ 2|V (F )| − 2 for every nonempty F ⊆ E \ L(E), and let us refer to the set of the counting conditions (L1), (L2’), (L3) and (L4) as the weak counting condition. Although the detailed description is omitted, Theorem 2 easily follows from the next result.
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(a)
(b)
Fig. 3. (a) A proper (k, 2)-rooted-forest partition for k = 3. (b) A non-proper (k, 2)rooted-forest partition for k = 3.
Theorem 3. Let Gc be a loop-colored graph, and let k be the number of colors used in Gc . Then, Gc satisfies the weak counting condition if and only if it admits a (k, 2)-rooted-forest partition. Notice that, a set of edges satisfying the weak counting condition and the strong counting condition, respectively, is a base of the matroid induced by the integervalued nondecreasing submodular functions μ : 2E → Z and μ : 2E → Z, respectively, defined as μ(F ) = 2|V (F )| − 2 + min{χ(F ), 2}
μ (F ) =
2|V (F )| − 3 2|V (F )| − 2 + min{χ(F ), 2}
if L(F ) = ∅ otherwise,
(4)
(5)
where χ(F ) denotes the total number of colors appearing in L(F ). Therefore, Theorem 3 implies that a set of edges admitting a (k, 2)-rooted-forest partition is characterized in terms of a matroid as the well-known characterization of forestpartitions in terms of the union of graphic matroids. Also, Theorem 2 generalizes Crapo’s characterization [1] of Laman graphs. Proof of Theorem 3 for k = 2. We now consider a special case of Theorem 3 where the number of colors k is restricted to two. We prove that (2, 2)-rootedforest partitions can be characterized in terms of the union of two matroids. For a vertex set V , let K(V ) denote the complete graph on V , and let K + (V ) denotes the graph obtained from K(V ) by attaching two loops at each vertex. We simply denote by K + (V ) the edge set of K + (V ), if it is clear from the context. Throughout this restricted case (of k = 2), we assume that one of two loops incident to v in K + (V ) is colored in red and the other one is colored in blue for each v ∈ V . For an edge set F ⊆ K + (V ), let Lr (F ) and Lb (F ) denote the sets of loops colored in red and blue, respectively. + Let us first consider the following function τr : 2K (V ) → Z; For F ⊆ K + (V ), τr (F ) = |V (F )| − 1 + χr (F ),
(6)
where χr (F ) is defined as χr (F ) = 0 if Lr (F ) = ∅ and otherwise χr (F ) = 1. Then, it is not difficult to see that τr is submodular. Also τr is nondecreasing,
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and hence it induces a matroid, denoted by Mτr , on K + (V ). The functions τb and χb , and the matroid Mτb (for the blue color) are symmetrically defined. Lemma 2. Let c ∈ {r, b} and c¯ ∈ {r, b} \ {c}. An edge set F of K + (V ) is a base of Mτc if and only if it is a spanning rooted-forest colored in c. Let us consider how the independent sets of Mτr ∨ Mτb can be characterized in terms of the counting condition. To apply Lemma 1, we just need to show the following property. Lemma 3. Let c ∈ {r, b} and let F ⊆ K + (V ). Let m be the total number of connected components of G[F ] and {F1 , . . . , Fm } be a partition of F such that, for each j = 1, . . . , m, Fj is the edge set of a connected component of G[F ]. Also, let F0 = ∅. Then, {F0 , F1 , . . . , Fm } ∈ P(F ) takes theminimum value of m (1). Namely, the rank rτc (F ) can be described by rτc (F ) = i=1 τc (Fi ). Combining Lemma 1 and Lemma 3, we obtain Mτr ∨ Mτb = Mτr +τb , and the following lemma easily follows. Lemma 4. Let E ⊆ K + (V ). Then, E is a base of Mτr ∨ Mτb if and only if it satisfies the weak counting condition. We are now ready to show Theorem 3 for k = 2. By Lemma 2, an edge set is a base of Mτr ∨ Mτb if and only if it can be partitioned into Er and Eb which are spanning rooted-forests colored in red and blue, respectively, and equivalently it admits a (2, 2)-rooted-forest partition. Combining Lemma 4 with this fact, we conclude that an edge set admits a (2, 2)-rooted-forest partition if and only if it satisfies the weak counting condition. Due to the space limitation, we omit the proof of Theorem 3 for k > 2 in this extended abstract. Other Combinatorial Results. In order to prove Theorem 1 we need one more combinatorial lemma related to the weak counting condition. We refer to a vertex v as a (a, b)-vertex in Gc if δE\L(E) (v) = a and δL(E) (v) = b. The following lemma claims the existence of small degree vertices. Lemma 5. Let Gc = (V, E, c) be a connected graph with |V | > 1 satisfying the weak counting condition. Suppose that there exists no (1, 1), (1, 2), (2, 0), (3, 0)vertex. Then, for any (k, 2)-rooted-forest partition E of Gc , there exists a (2, 1)vertex v such that the two non-loop edges incident to v belong to distinct colored classes in E. Remark. Each condition of the strong counting condition (and also the weak one) can be checked by using the so-called “pebble game” algorithm for checking the sparsity of graphs (see [5,13]). Since the pebble game works in O(|V |2 ) time, checking whether Gc satisfies the strong counting condition (also the weak one) can be done in O(k|V |2 ) time, where k is the number of colors. This can be improved to O(|V |2 ) time by just avoiding to start each pebble game from scratch. Therefore, assuming a generic joint configuration, we can decide in O(|V |2 ) time whether a bar-slider framework is rigid or not.
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Infinitesimally Rigid Bar-Slider Frameworks
We now prove the following statement; for a given loop-colored graph Gc satisfying the strong counting condition and a direction mapping d on a set of colors, there exists a joint configuration p such that the bar-slider framework (Gc , p, d) is infinitesimally rigid in the plane. Note that, if one particular realization (Gc , p, d) is rigid, then (Gc , q, d) becomes rigid for all generic joint configurations q as explained in Section 2, implying the nontrivial part (sufficiency) of Theorem 1. The proof is done by induction on |E \ L(E)| as follows. We convert Gc to a slightly smaller graph Gc with respect to |E \ L(E)|, and then prepare a rigid realization (Gc , p, d) based on the induction hypothesis. Finally, we shall show that (Gc , p, d) is infinitesimally rigid. We omit the base case. Let us consider the case of |E \ L(E)| > 0. Applying Theorem 2, we have a proper (k, 2)-rooted-forest partition E = {E1 , E2 , . . . , Ek } of Gc , where k = χ(E). For the convenience of the description, let us assume that all edges of Gc (not only loops but also non-loop edges) are colored according to this partition E throughout the proof. Following Lemma 5, the proof is split into five cases depending on the existence of small degree vertices. Due to the space limitation, we shall omit the four cases, and show only the final case where Gc has no (1, 1), (1, 2), (2, 0), (3, 0)-vertex. In this case, by Lemma 5, Gc has a (2, 1)-vertex v such that the two non-loop edges incident to v belong to distinct colored classes in E. Let e1 be the loop attached to v with the color c1 , and let va and vb be the two non-loop edges incident to v. By condition (P2) of E and Lemma 5, we may assume that va and vb are colored in c1 and c2 , respectively (see Figure 4). We shall consider the graph Gc obtained from Gc by removing va, vb and then attaching new loops f1 colored in c1 to a and f2 colored in c2 to v, respectively (see Figure 4). Then, the coloring of the edge set induces a proper (k, 2)-rootedforest partition of Gc , implying that Gc satisfies the strong counting condition. By induction, there exists a rigid realization (Gc , p, d). Since the joint p(v) is isolated, we may assume that p(v) is located such that p(v) − p(a) is orthogonal to the direction d(c1 ) (as shown in Figure 4). We claim that (Gc , p, d) is infinitesimally rigid. To see this, suppose that there exists a nonzero infinitesimal motion v for (Gc , p, d). Due to the direction constraint by the slider associated with e1 , v(v) is a scalar multiple of d(c1 ), and consequently v(a) is also a scalar multiple of d(c1 ) by the length constraint of the bar p(v)p(a). Let us define v : V → R2 as v (v) = 0 and v (u) = v(u) for
e1
e1
v b
a
f1
p(v)
v f2
a
Gc
G’c’
realization
b
d(c1)
p(v)
d(c1) p(a)
p(b)
(G’c’ ,p,d)
Fig. 4. (2, 1)-vertex
p(b) p(a)
(Gc ,p,d)
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all u ∈ V \ {v}. Then, it is not difficult to see that v satisfies all the constraints appearing in (Gc , p, d) since v satisfies the direction constraints by the sliders associated with f1 and f2 , only which are not contained in Gc . Therefore, all entries of v must be zero because (Gc , p, d) is rigid. Since v is nonzero, we obtain that v(v)
= 0 and v(u) = v (u) = 0 for all u ∈ V \ {v}. However, v does not satisfy the length constraint (2) by the bar p(v)p(b) which (Gc , p, d) has. This contradicts that v is an infinitesimal motion of (Gc , p, d).
References 1. Crapo, H.: On the generic rigidity of plane frameworks. Technical Report 1278, Institute de recherche d’informatique et d’automatique (1988) 2. Fekete, Z.: Source location with rigidity and tree packing requirements. Operations Research Letters 34(6), 607–612 (2006) 3. Fekete, Z., Jord´ an, T.: Uniquely localizable networks with few anchors. In: Nikoletseas, S.E., Rolim, J.D.P. (eds.) ALGOSENSORS 2006. LNCS, vol. 4240, pp. 176–183. Springer, Heidelberg (2006) 4. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4, 331–340 (1970) 5. Lee, A., Streinu, I.: Pebble game algorithms and sparse graphs. Discrete Mathematics 308(8), 1425–1437 (2008) 6. Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proc. 19th Canadian Conference on Computational Geometry, CCCG 2007 (2007) 7. Lov´ asz, L.: Matroid matching and some applications. J. Combinatorial Theory, Series (B) 28, 208–236 (1980) 8. Oxley, J.: Matroid Theory. Oxford University Press, Oxford (1992) 9. Pym, J.S., Perfect, H.: Submodular functions and independence structures. J. Math. Analysis Appl. 30, 1–31 (1970) 10. Servatius, B., Shai, O., Whiteley, W.: Combinatorial characterization of the assur graphs from engineering. arXiv:0801.252v1 (2008) 11. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003) 12. Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: Proc. ACM Symposium on Computational Geometry (SoCG 2008), pp. 365–374 (2008) 13. Streinu, I., Theran, L.: Sparsity-certifying graph decompositions. Graphs and Combinatorics 25(2), 219–238 (2009) 14. Whiteley, W.: Matroids from discrete geometry. In: Bonin, J., Oxley, J., Servatius, B. (eds.) Matroid Theory. AMS Contemporary Mathematics, pp. 171–313 (1997)
