A characterization of chain probe graphs

A chain probe graph is a graph that admits an independent set S of vertices and a set F of pairs of elements of S such that G+ F is a chain graph (i.e., a 2K2-free bipartite graph). We show that chain probe graphs are exactly the bipartite graphs that do not contain as an induced subgraph a member of a family of six forbidden subgraphs, and deduce an O(n2) recognition algorithm. Given a class C of graphs, a graph G= (V ,E) is said to be a C probe graph if there exists an independent set of vertices S ⊆ V and a set F of pairs of vertices of S such that the graph G′ = (V ,E ∪ F) is in class C. The C probe Graph Recognition problem is a special case of the C-Graph Sandwich problem (Golumbic et al. 1995) when S is given as part of the input. Probe graph problems have been investigated on various families of graphs (Chandler et al. 2008; Chang et al. 2005; Le and de Ridder 2007a, 2007b; Przulj and Corneil 2005). Specifically, interval probe graphs were introduced by Zhang (1994) for applications in biology, and they were studied further in (Golumbic and Trenk 2004; Johnson and Spinrad 2001; McConnell and Spinrad 2002; McMorris et al. 1998; Sheng 1999; Zhang et al. 1999a, 1999b). Chordal probe graphs were introduced by Golumbic and Lipshteyn (Golumbic and Lipshteyn 2003; Golumbic and Lipshteyn 2004) and studied by Berry et al. (2004, 2007). Here we consider the class of so-called chain graphs. Recall that a 2K2 is a graph with four vertices a, b, c, d and two edges ab, cd . Given a graph H , a graph G is H -free if G does not have an induced subgraph that is isomorphic to H . The 2K2-free bipartite graphs This research was made possible by the joint French–Israeli project Recognition, Decomposition, and Optimization Problems in Graph Theory. M.C. Golumbic Caesarea Rothschild Institute and Department of Computer Science, University of Haifa, Haifa, Israel F. Maffray ( ) C.N.R.S., Laboratoire G-SCOP, Grenoble, France e-mail: [email protected] G. Morel Laboratoire G-SCOP, Grenoble, France 176 Ann Oper Res (2011) 188:175–183 are known in literature under the name chain graphs (Yannakakis 1981) or difference graphs (Hammer et al. 1990). There is a natural connection between chain graphs and the famous class of threshold graphs introduced by Chvátal and Hammer (1977). A graph is a threshold graph if its vertexset can be partitioned into a clique Q and an independent set R such that any two vertices in R have inclusionwise comparable neighbourhoods. It is easy to see that a bipartite graph G = (A,B;E) is a chain graph if and only if the graph obtained from G by making all vertices in A pairwise adjacent is a threshold graph. Conversely, a graph H whose vertexset can be partitioned into a clique Q and an independent set R is a threshold graph if and only if the bipartite graph obtained from H by removing all edges in Q is a chain graph. Threshold graphs have been the subject of many articles and of a book (Mahadev and Peled 1995) by Mahadev and Peled, two of Peter Hammer’s former doctoral students. The main characterizing property of chain graphs (see Mahadev and Peled 1995) is that their vertices, in each side of the bipartition, can be ordered under inclusion of their neighbourhood. More formally: Proposition 1 Let G= (V1,V2,E) be a connected bipartite graph. Then the following conditions are equivalent: – G is 2K2-free; – There exists a total ordering on V1 such that any two vertices v1,w1 ∈ V1 satisfy v1 w1 if and only if N(v1)⊇N(w1); – There exists a total ordering on V2 such that any two vertices v2,w2 ∈ V2, satisfy v2 w2 if and only if N(v2)⊇N(w2). Any ordering on the vertices of V1 or V2 that satisfies the above condition may be called a chain ordering. Definition 1 A chain probe graph is any bipartite graph G= (V ,E) such that there exists an independent set of vertices S ⊆ V and a set F of pairs of vertices of S such that the graph G′ = (V ,E ∪ F) is 2K2-free. It will be convenient to denote by G+ F the graph (V ,E ∪ F). Note that if G is a chain probe graph then so is every induced subgraph of G. In other words the class of chain probe graphs is hereditary for induced subgraphs, and consequently there is a list of minimally non-chain probe graphs. Our goal here is to characterize chain probe graphs, that is, to determine that list explicitly. First, we make the following observation, which will be useful in the continuation. We denote by Pk the chordless path with k vertices, by Ck the chordless cycle with k vertices, and by kK2 the graph with 2k vertices and k disjoint edges. Lemma 1 Suppose that G is a chain probe graph with sets S and F as in Definition 1. Then: 1. If G contains a P5 v1-v2-v3-v4-v5, then S contains either v1, v4 or v2, v5 and no other vertex of the P5, and correspondingly F contains either v1v4 or v2v5. 2. If G contains a P6 v1-v2-v3-v4-v5-v6, then S contains v2, v5 and no other vertex of the P6 and F contains v2v5. Ann Oper Res (2011) 188:175–183 177 Fig. 1 Graphs (a) H1, (b) H2 and (c) H3 Proof First suppose that G contains a P5 v1-v2-v3-v4-v5. Then F must contain one of the edges v1v4, v2v5, for otherwise v1, v2, v4, v5 will still induce a 2K2 in G + F . So S must contain v1, v4 or v2, v5; and in either case, the fact that S is an independent set implies that S contains no other vertex of the P5. Now suppose that G contains a P6 v1-v2-v3-v4-v5-v6. Then the preceding argument can be applied to the two P5’s v1-· · ·-v5 and v2-· · ·-v6, and since S must be an independent set, it is easy to see that the only possibility is that S contains v2, v5 and no other vertex of the P6 and F contains v2v5. Not all bipartite graphs are chain probe, as shown in Lemma 2: Lemma 2 Graphs C6, P7, 3K2 and the three graphs denoted by H1, H2 and H3 (see Fig. 1) are not chain probe graphs. Proof Let G be one of these six graphs. Suppose that G is a chain probe graph with sets S and F as in the definition. Since G contains a 2K2, we have F = ∅ and |S| ≥ 2. First let G be a C6, with vertices v1, . . . , v6 and edges vivi+1 modulo 6. Since F = ∅ and |S| ≥ 2, we may assume without loss of generality that v1, v4 are in S and v1v4 is in F . Then, no other vertex of G is in S because S is an independent set. But then G + F still contains the 2K2 with edges v2v3 and v5v6. Now let G be a P7 v1-· · ·-v7. We can apply Part 2 of Lemma 1 to the P6 v1-· · ·-v6, which implies that v2, v5 are in S, and to the P6 v2 · · ·v7, which implies that v3, v6 are in S. But this contradicts the fact that S is independent. Now let G be a 3K2 with vertices v1, v2, v3,w1,w2,w3 and edges v1w1, v2w2 and v3w3. Since S is independent, for each i ∈ {1,2,3}, S cannot contain both vi,wi . So up to symmetry we may assume that it does not contain the wi ’s. Consequently, and since G+ F must be bipartite, F cannot contain all three pairs consisting of two vi ’s, say F does not contain v1v3. But then v1,w1, v3,w3 still induce a 2K2 in G+ F . Now let G be any of H1,H2,H3, with vertices a, b, c, d, a′, b′, c′, d ′ and edges ab, bc, cd, a′b′, b′c′, c′d ′, bb′, cc′ plus any subset of {aa′, dd ′}. We can apply Part 1 of Lemma 1 to the P5 a-b-c-c′-d ′, which implies that one of b, c′ is in S, and to the P5 a′-b′-c′c-d , which implies that one of b′, c is in S. But this contradicts the fact that S is independent. Thus the lemma holds. The following theorem proves that these six graphs are the only minimal non-chain probe graphs. In a graph G= (V ,E), we say that a vertex v ∈ V is complete to a set X ⊂ V if v is adjacent to every vertex of X. We say that v ∈ V is anti-complete to X ⊂ V if v is not adjacent to any vertex of X. We say that a set Y is complete (respectively, anti-complete) to X if every vertex of Y is complete (respectively, anti-complete) to X. Given two disjoint sets A,B , let us denote by A⊗B the set of all unordered pairs {a, b} with a ∈A and b ∈ B . 178 Ann Oper Res (2011) 188:175–183 Theorem 1 Let G be a bipartite graph. Then G is a chain probe graph if and only if it is {C6,P7,3K2,H1,H2,H3}-free. Proof The necessary condition comes directly from Lemma 2. To prove the sufficient condition, we shall construct the graph step by step. So let G be any a bipartite {C6,P7,3K2,H1,H2,H3}-free graph. We distinguish between three cases. (I) First case: suppose G has an induced P6. Then there are six pairwise disjoint nonempty independent sets X1, . . . ,X6 such that for i = 1, . . . ,5, every vertex in Xi has a neighbour in Xi+1 and every vertex in Xi+1 has a neighbour in Xi , and for every i, j ∈ {1, . . . ,6} with |j − i| ≥ 2 there is no edge between Xi and Xj . Let X =X1 ∪ · · · ∪X6. We choose the 6-tuple (X1, . . . ,X6) so that the set X3∪X4 is maximal and, under this condition, the set X is maximal. We claim that: X1 ∪X2 and X5 ∪X6 induce 2K2-free graphs. (1) For suppose on the contrary, and up to symmetry, that there is a 2K2 induced by some vertices x1,w1 ∈ X1, x2,w2 ∈ X2, with edges x1x2, w1w2. By the definition of the Xi ’s, there is an edge x5x6 with x5 ∈X5, x6 ∈X6. Then G contains a 3K2 with edges x1x2, w1w2 and x5x6, a contradiction. Thus (1) holds. By (1) and Proposition 1, there exists a vertex v1 ∈ X1 that is complete to X2, a vertex v2 ∈X2 that is complete to X1, a vertex v5 ∈X5 that is complete to X6 and a vertex v6 ∈X6 that is complete to X5. X2 is complete to X3, and X4 is complete to X5. (2) For suppose on the contrary, and up to symmetry, that there exist non-adjacent vertices w2 ∈ X2 and w3 ∈ X3. By the definition of the Xi ’s, w3 has a neighbour x2 ∈ X2 and a neighbour x4 in X4; and x4 has a neighbour x5 ∈ X5, and by (1) x5 is adjacent to v6. But then w2-v1-x2-w3-x4-x5-v6 is an induced P7 in G, a contradiction. Thus (2) holds. X3 ∪X4 induces a 2K2-free graph. (3) For suppose that some vertices v3,w3 ∈X3 and v4,w4 ∈X4 induce a 2K2 with edges v3v4 and w3w4. Then v2,w3,w4, v5, v4, v3 induce a C6, a contradiction. Thus (3) holds. By (3) and Proposition 1, there exists a vertex v3 ∈X3 that is complete to X4 and a vertex v4 ∈X4 that is complete to X3. Note that, by (1)–(3), the set X induces a connected subgraph of G. Moreover, for each vertex x1 ∈ X1, G contains an induced P6 x1-v2-v3-v4-v5-v6, and a similar property holds for X2, . . . ,X6. Now we define subsets of V (G) \X as follows. Y2 = {v / ∈X | v has a neighbour in X3 and is anti-complete to X \X3}. Y5 = {v / ∈X | v has a neighbour in X4 and is anti-complete to X \X4}. Z2 = {v / ∈X | v is complete to X1 ∪X3 ∪X5}. Z5 = {v / ∈X | v is complete to X2 ∪X4 ∪X6}. T = {v / ∈X | v is anti-complete to X}.