- Email: [email protected]
On the bi-enhancement of chordal-bipartite probe graphs
Information Processing Letters 110 (2010) 193–197
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
On the bi-enhancement of chordal-bipartite probe graphs Elad Cohen a,∗ , Martin Charles Golumbic a , Marina Lipshteyn a , Michal Stern a,b a b
Caesarea Rothschild Institute, and Department of Computer Science, University of Haifa, Haifa, Israel Academic College of Tel-Aviv-Jaffa, Israel
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 17 March 2009 Received in revised form 11 November 2009 Accepted 7 December 2009 Available online 11 December 2009 Communicated by S.E. Hambrusch Keywords: Combinatorial problems Chordal bipartite graphs Probe graphs
Given a class C of graphs, a graph G = ( V , E ) is said to be a C -probe graph if there exists a stable (i.e., independent) set of vertices X ⊆ V and a set F of pairs of vertices of X such that the graph G = ( V , E ∪ F ) is in the class C . Recently, there has been increasing interest and research on a variety of C -probe graph classes, such as interval probe graphs, chordal probe graphs and chain probe graphs. In this paper we focus on chordal-bipartite probe graphs. We prove a structural result that if B is a bipartite graph with no chordless cycle of length strictly greater than 6, then B is chordal-bipartite probe if and only if a certain “enhanced” graph B ∗ is a chordal-bipartite graph. This theorem is analogous to a result on interval probe graphs in Zhang (1994) [18] and to one on chordal probe graphs in Golumbic and Lipshteyn (2004) [11]. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Let G = ( V , E ) be a finite, undirected, simple graph. We say that vertex u “sees” vertex v if (u , v ) ∈ E. A set X ⊆ V is a stable set in G if for all u , v ∈ X , (u , v ) ∈ / E, i.e., no vertex in X sees another vertex in X . A graph G is a bipartite graph if its vertices can be partitioned into two disjoint stable sets V = X ∪ Y . We will refer to this as the “black/white” bipartition of the vertices. A sequence ( v 1 , . . . , v n ) of distinct vertices is a path in G if ( v i , v i +1 ) ∈ E for 1 i n − 1. A closed path ( v 1 , . . . , v n , v 1 ) is called a cycle if in addition ( v n , v 1 ) ∈ E. A chord of a cycle ( v 1 , . . . , v n , v 1 ) is an edge between two vertices of the cycle that is not an edge of the cycle. A chordless cycle C n is a cycle which contains no chords and has n vertices and n edges. A graph G is a chordal graph if it contains no induced chordless cycle C n , for n 4. A graph G is a chordalbipartite graph if it is a bipartite graph and it contains no induced chordless cycle C n , for n 6, as introduced in [8]. (Note that just as a chordal graph may contain triangles, a chordal-bipartite graph may contain chordless 4-cycles.)
*
Corresponding author. E-mail address: [email protected] (E. Cohen).
0020-0190/$ – see front matter doi:10.1016/j.ipl.2009.12.003
© 2009 Elsevier B.V.
All rights reserved.
Let C be a graph class. A graph G = ( V , E ) is called a C -probe graph if its vertex set can be partitioned into two subsets, P (probes) and N (non-probes), where N is a stable set, and one can add to G a set of edges between pairs of non-probes such that the resulting graph is in C . More formally, G is C -probe if there exists a completion F ⊆ {(u , v ) | u , v ∈ N , u = v } such that G = ( V , E ∪ F ) is in C . Recognizing whether or not G is a C -probe graph is known as the non-partitioned probe problem. In the partitioned version of the probe problem, the partition of V into probes and non-probes is given and fixed as part of the input. The partitioned C -probe problem is a special case of the C -sandwich problem [3,9,14]. Interval probe graphs were first introduced by Zhang [18] and studied further in [10,15–17], see also [13]. Chordal probe graphs were investigated in [11] and a characterization and recognition algorithm was given in [2] for both the partitioned and non-partitioned versions. Recently, characterizations and recognitions have been given for chain probe graphs in [12], and for threshold probe graphs and trivially-perfect probe graphs, also known as quasi-threshold probe graphs, in [4]. Other probe classes are to be found in [5,6]. In this paper, we focus on the partitioned version of chordal-bipartite probe graphs. In particular, we prove a
194
E. Cohen et al. / Information Processing Letters 110 (2010) 193–197
structural result that if B is a bipartite graph with no chordless cycle of length strictly greater than 6, then B is chordal-bipartite probe if and only if a certain “enhanced” graph B ∗ is a chordal-bipartite graph. This theorem is analogous to a result on interval probe graphs in [18] and to one on chordal probe graphs in [11]. We believe it may also shed light on the more general case of the partitioned version, where B may have chordless cycles of length larger than 6. We conclude the paper with open questions. All standard definitions can be found in [7]. 2. Motivation: chordal probe graphs Let G = ( V , E ) be a graph whose vertices have been partitioned into a set P of probes and a stable set N of non-probes. The following was proved in [11]: Lemma 2.1. (See [11].) If G is a chordal probe graph with respect to the partition P ∪ N, then probes and non-probes must alternate on every chordless cycle. In the specific case of a chordless 4-cycle with edges ab, bc , cd, da, this means that either {a, c } are probes and {b, d} are non-probes, or vice versa. Moreover, suppose that {a, c } are probes, then any possible chordal probe completion of G would be forced to contain the addition of an edge bd, which Zhang [18] called an enhanced edge. The enhanced graph G ∗ = ( P ∪ N , E ∗ ) is defined to be the graph obtained from G by adding all enhanced edges from all chordless 4-cycles. Theorem 2.2. (See [11].) Let G be a graph containing no induced chordless cycles C k for k > 4. If G has a probe/non-probe partition in which probes and non-probes alternate on every chordless 4-cycle, then the enhanced graph G ∗ is a chordal completion of G. Lemma 2.1 together with Theorem 2.2 prove the next corollary.
of the vertices and the “probe/non-probe” bipartition of the vertices are generally different! Moreover, if B is connected, then the completion edges between non-probes will have one endpoint white and the other black, in order to maintain the bipartite property. We begin by stating some necessary properties due to Berry, et al. [1]. The reader may wish to reconstruct a proof. Lemma 3.1. If B is a chordal-bipartite probe graph with respect to the partition P ∪ N, then on every chordless cycle of length 6 in B the following must hold: (1) every probe sees at most one other probe, (2) there is at least one edge of the cycle whose endpoints are both probes. Lemma 3.1 implies, in particular, that (1) on a chordless cycle there is no consecutive triple of probes, and that (2) there must be at least two pairs of consecutive probes, due to the parity of a bipartite graph. Moreover, Remark 3.2. In the specific case of a chordless 6-cycle C 6 , there are exactly 2 non-probes, one white and one black, opposite each other, and any possible chordal-bipartite probe completion of B must contain the added edge joining them. We will call this forced edge a bi-enhanced edge. Thus, with respect to a given probe/non-probe partition, we define the bi-enhanced graph B ∗ to be the graph obtained from B by adding all bi-enhanced edges from all chordless 6-cycles. We are now ready to state our main result. Theorem 3.3. Let B be a bipartite graph that contains no chordless cycle C k for k > 6. If B has a probe/non-probe partition in which probes and non-probes satisfy the property in Remark 3.2 on every chordless 6-cycle, then B ∗ is a chordal-bipartite completion of B.
Corollary 2.3. If G is a C k -free graph for k > 4, then G is chordal probe if and only if G ∗ is chordal.
Similar to the case of chordal probe graphs, Lemma 3.1 together with Theorem 3.3 immediately prove the next corollary.
This also gave an alternate proof of a result of Zhang [18].
Corollary 3.4. If B is a C k -free bipartite graph for k > 6, then B is chordal-bipartite probe if and only if B ∗ is chordal-bipartite.
Corollary 2.4. If G is an interval probe graph, then G ∗ is chordal.
See Fig. 1 for an example of a completed chordalbipartite probe graph with all its bi-enhanced edges.
3. Chordal-bipartite probe graphs We now turn our attention to the probe problem for chordal-bipartite graphs. Our goal will be to obtain a result in the same spirit as Corollary 2.3 in the case of a bipartite graph that has no chordless cycles of size strictly greater than 6. Let B = ( P ∪ N , E ) be a bipartite graph whose vertex set is partitioned into P (probes) and N (non-probes), where N is always assumed to be a stable set. Note that in the case of a bipartite graph B, the “black/white” bipartition
4. Proof of Theorem 3.3 This section is devoted to the proof of Theorem 3.3. Proof. Clearly, B ∗ is a bipartite graph. We will prove that B ∗ has no chordless cycle of size greater than 4. Suppose B ∗ has a chordless cycle of size greater than 4 and let h be the minimal number of bi-enhanced edges in such a cycle. Let C = (x1 , . . . , xm , x1 ) be a chordless cycle in B ∗ with h bi-enhanced edges.
E. Cohen et al. / Information Processing Letters 110 (2010) 193–197
195
on C and ai has exactly one neighbor on C , namely x1 , then b i is adjacent to all vertices of opposite color on C .
Fig. 1. Example of a completion of a chordal bipartite probe graph. Black and white vertices correspond to the bipartition sets.
If h = 0, then C is also a chordless cycle in B, and therefore C is of size 6. By Remark 3.2, C must have a chord which is a bi-enhanced edge. Contradiction! Thus, we may assume that h > 0. Without loss of generality, we may assume that (x1 , x2 ) is a bi-enhanced edge in C , and that (x1 , x2 ) was added in the construction of B ∗ as the chord of a chordless cycle C = (x1 , a1 , b1 , x2 , b2 , a2 , x1 ) in B of size 6. Claim 4.1. Suppose ai ∈ / C , for i = 1 or i = 2, then either ai has exactly one neighbor on C , namely x1 , or ai is also adjacent to xm−1 (see Fig. 2). Respectively, suppose b i ∈ / C , then either bi has exactly one neighbor on C , namely x2 , or b i is also adjacent to x4 .
Proof. We prove the first case only, since the second case is similar. Suppose there exists xl ∈ C that is of opposite color, but is not adjacent to ai , such that l is maximal. By Claim 4.1, ai is adjacent to xm−1 and thus l < m − 1. Suppose none of xl−1 , . . . , x3 is adjacent to ai . Let C = (ai , xl+2 , . . . , x3 , x2 , bi , ai ). Since (ai , xl+2 ) ∈ E ( B ), the cycle C is chordless cycle of size greater than 4 with at most h − 1 bi-enhanced edges. Contradiction! Otherwise, let xk ∈ C be a neighbor of ai , such that k is maximal and k < l. Since (ai , xk ) ∈ E ( B ), the cycle (ai , xl+2 , . . . , xk , ai ) is a chordless cycle of size greater than 4 with at most h − 1 bi-enhanced edges. Contradiction! 2 Claim 4.3. At most one of a1 , a2 , b1 , b2 is on C . Proof. Without loss of generality, suppose that a1 ∈ C . It immediately follows that a1 = xm , since C is chordless / C , since otherand (x1 , a1 ) ∈ E ( B ). Then obviously a2 ∈ / C , since otherwise wise a2 = a1 = xm . In addition, b1 ∈ (a1 , b1 ) is a chord in C . Suppose that b2 ∈ C , meaning that b2 = x3 . The cycle (x1 , a2 , b2 = x3 , x4 , . . . , xm = a1 , x1 ) is not chordless in B ∗ , since otherwise it has at most h − 1 bi-enhanced edges, contradicting the minimality of C . Therefore, a2 has more than one neighbor on C . Thus, by Claim 4.1, a2 is adjacent to xm−1 . Hence, (xm−1 , a2 , b2 = x3 , x2 , b1 , a1 = xm , xm−1 ) is a chordless cycle of size greater than 4 in B ∗ with at most h − 1 bi-enhanced edges, contradicting the minimality of C . 2
Proof. Suppose ai has more than one neighbor on C . Let x j ∈ C be the neighbor of ai , such that j is maximal. Then C = (ai , x1 , xm , . . . , x j , ai ) is a chordless cycle with at most h − 1 bi-enhanced edges, since (ai , x j ) ∈ E ( B ). By minimality of h in C , C must be of size 4 ( j = m − 1) and xm−1 is adjacent to ai . By symmetry, either b i has exactly one neighbor on C or b i is adjacent to x4 . 2
Definition 4.4 (bypass). Let (x1 , x2 ) be a bi-enhanced edge on C in B ∗ . Let [x1 , a, b, x2 ] be a chordless path in B, where a, b are probes. The pair a, b is called a bypass of / C and the edge (x1 , x2 ) with respect to C in B ∗ if a, b ∈ each of a, b has exactly one neighbor on C , namely, x1 and x2 , respectively.
Claim 4.2. Suppose V (C ) ∩ V (C ) = {x1 , x2 }. If ai has more than one neighbor on C and b i has exactly one neighbor on C , namely x2 , then ai is adjacent to all vertices of opposite color on C (see Fig. 2). Respectively, if b i has more than one neighbor
Proof. We prove that the bi-enhanced edge (x1 , x2 ) has a bypass. By Claim 4.3, at most one of a1 , a2 , b1 , b2 is on C . Therefore, there are two cases:
Claim 4.5. Every bi-enhanced edge on C in B ∗ has a bypass.
Case (i): Exactly one of a1 , b1 , a2 , b2 is on C .
Fig. 2. Examples of graphs B ∗ : (a) example where h = 1, under the conditions described in Claim 4.1; (b) example where h > 1 and some of the edges (xi , xi +1 ) are bi-enhanced, under the conditions described in Claim 4.2.
196
E. Cohen et al. / Information Processing Letters 110 (2010) 193–197
Fig. 3. Example of a graph B ∗ under the conditions described in Case (i).
Without loss of generality, suppose that a2 ∈ C , meaning that a2 = xm . Suppose a1 has more than one neighbor on C . Then, by Claim 4.1, a1 is adjacent to xm−1 . Consider the cycle C = (a2 = xm , xm−1 , a1 , b1 , x2 , b2 , a2 ). Due to the bipartition (xm−1 , b2 ), (xm−1 , b1 ) ∈ / E ( B ∗ ). Since a1 , a2 are probes, (xm−1 , a1 ), (xm−1 , a2 ) ∈ E ( B ). Then C is a chordless cycle of size 6 in B with no bi-enhanced edges, a contradiction. Therefore, a1 has exactly one neighbor on C . Suppose b1 has more than one neighbor on C . Let xi be a neighbor of b1 , such that i is maximal. Then, the cycle (a2 = xm , . . . , xi , b1 , a1 , x1 , a2 ) is chordless cycle of size greater than 4 in B ∗ with at most h − 1 bi-enhanced edges, contradicting the minimality of C . Therefore, b1 has only one neighbor on C . Thus, a1 , b1 is a bypass. Case (ii): None of a1 , b1 , a2 , b2 is on C . If a1 and b1 , and/or a2 and b2 have only one neighbor on C , then a1 , b1 and/or a2 , b2 are bypasses. Assume this is not the case. Suppose a1 and a2 both have more than one neighbor on C . Then, by Claim 4.1, a1 and a2 are adjacent / E ( B ∗ ), the cycle to xm−1 . Since (xm−1 , b1 ), (xm−1 , b2 ) ∈ (a1 , xm−1 , a2 , b2 , x2 , b1 , a1 ) is a chordless cycle of size 6 in B with no bi-enhanced edges. Contradiction! Thus, we may assume that a1 has only one neighbor on C . Similarly, at most one of b1 and b2 have more than one neighbor on C . Suppose b1 has more than one neighbor on C . Then, by Claim 4.1, (b1 , x4 ) ∈ E ( B ∗ ). Since, in this case, b2 has only one neighbor on C , then by Claim 4.2, (a2 , x3 ) ∈ E ( B ∗ ). Since (x3 , b1 ) ∈ / E ( B ∗ ) due to the bipartition, the cycle (a2 , x1 , a1 , b1 , x4 , x3 , a2 ) is a chordless cycle of size 6 in B ∗ with at most h − 1 bi-enhanced edges. Contradiction! Thus, b1 has only one neighbor on C , and a1 , b1 is a bypass. 2 Claim 4.6. A vertex cannot be in bypasses of different bienhanced edges in B ∗ . Proof. Suppose there exists a vertex u that participates in two bypasses a, b and a , b of two bi-enhanced edges (xi , xi +1 ) and (x j , x j +1 ) in C , respectively, such that i < j. Since u has only one neighbor in C , the two edges are consecutive, meaning that xi +1 = x j and u = b = a . Therefore, (xi , xi −1 , . . . , x j +1 , b , b, a, xi ) is a chordless cycle with
at most h − 2 bi-enhanced edges, contradicting the minimality of C . 2 We are now ready to complete the proof of the theorem. Let C be a cycle in B obtained from C , by substituting each bi-enhanced edge with the three edges of a bypass of that edge, due to Claim 4.6. Suppose C is a chordless cycle in B. Since h > 0 and C is a cycle in B ∗ with at least 6 vertices, the cycle C has at least 8 vertices. This is a contradiction, since B has no chordless cycle of size greater than 6. Therefore, C has a chord in B. Since C is chordless, and since each vertex in a bypass has only one neighbor on C , all such chords of C connect only bypass vertices of different bi-enhanced edges. Therefore, the theorem holds for h = 1 and we may now assume that h 2. Definition 4.7 (bridge). Let (xi , xi +1 ) and (x j , x j +1 ) be bienhanced edges on C in B ∗ with bypasses a, b and a , b , respectively, such that i < j. An edge between the two bypasses a, b and a , b is called a bridge. Notice that all the chords of C are bridges, and there may be at most two bridges between any pair a, b and a , b , since the graph is bipartite. There are two cases: Case (i): (a, b ) or (a , b) or both are bridges. Suppose (a, b ) is a bridge. Since C is a chordless cycle of size at least 6, at least one of the cycles (xi , a, b , a , x j +1 , . . . , xi ), (xi +1 , b, a, b , x j , . . . , xi +1 ) is a chordless cycle of size greater than 4 with at most h − 2 bi-enhanced edges, a contradiction (see Fig. 3). The case where (a , b) is a bridge is similar. Case (ii): (a, a ) or (b, b ) or both are bridges. Suppose (a, a ) is a bridge and (b, b ) is not a bridge. Since C is a chordless cycle of size at least 6, at least one of the cycles (xi , a, a , x j +1 , . . . , xi ), (xi +1 , . . . , x j , b , a , a, b, xi +1 ) is a chordless cycle of size greater than 4 with at most h − 2 bi-enhanced edges, a contradiction. The case where only (b, b ) is a bridge is similar. Suppose both (a, a ) and (b, b ) are bridges. Since C is a chordless cycle of size at least 6, at least one of the cycles (xi , a, a , x j +1 , . . . , xi ), (xi +1 , . . . , x j , b , b, xi +1 ) is a chordless cycle of size greater than 4 with at most h − 2 bienhanced edges, a contradiction.
E. Cohen et al. / Information Processing Letters 110 (2010) 193–197
Thus, there is no chordless cycle C of size greater than 4 in B ∗ . Hence, B ∗ is a chordal-bipartite graph. 2 5. Open questions In this paper, we prove that the properties of Lemma 3.1 characterize chordal-bipartite probe graphs in the case where the given graph contains no chordless cycles of size greater than 6. Berry et al. [1] have asked, what further conditions may be needed for these properties to characterize chordal-bipartite probe graphs in the general case? On the algorithmic side, given a probe/non-probe partition of a bipartite graph B, how do we most efficiently find the bi-enhanced edges, that is, build B ∗ ? What is the complexity of recognition of chordal-bipartite probe graphs, even in the case where there are no chordless cycles of size greater than 6? References [1] A. Berry, E. Cohen, M.C. Golumbic, M. Lipshteyn, N. Pinet, A. Sigayret, M. Stern, Recognizing chordal-bipartite probe graphs, France–Israel binational science collaboration project, unpublished technical report, 2007. [2] A. Berry, M.C. Golumbic, M. Lipshteyn, Recognizing chordal probe graphs and cycle-bicolorable graphs, SIAM Journal on Discrete Mathematics 21 (2007) 573–591. [3] C. Bornstein, C.M.H. de Figueiredo, V.G.P. de Sá, The pair completion algorithm for the homogeneous set sandwich problem, Information Processing Letters 98 (3) (2006) 87–91. [4] D. Bayera, V.B. Leb, H.N. de Ridder, Probe threshold and probe trivially perfect graphs, Theoretical Computer Science 410 (47–49) (2009) 4812–4822.
197
[5] D.B. Chandler, M.S. Chang, T. Kloks, S.L. Peng, Probe Graphs, preliminary version, 2009. [6] M.S. Chang, T. Kloks, D. Kratsch, J. Liu, S.-L. Peng, On the recognition of probe graphs of some self-complementary classes of perfect graphs, LNCS 3595 (2005) 808–817. [7] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980; M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, second edition, Annals of Discrete Mathematics, vol. 57, Elsevier, Amsterdam, 2004. [8] M.C. Golumbic, C.F. Goss, Perfect elimination and chordal-bipartite graphs, Journal of Graph Theory 2 (1978) 155–163. [9] M.C. Golumbic, H. Kaplan, R. Shamir, Graph sandwich problems, Journal of Algorithms 19 (1995) 449–473. [10] M.C. Golumbic, M. Lipshteyn, On the hierarchy of interval, probe and tolerance graphs, in: Proc. 32th Southeastern Conf. on Combinatorics, Graph Theory and Computing, in: Congressus Numerantium, vol. 153, Utilitas Math., Winnipeg, Canada, 2001, pp. 97–106. [11] M.C. Golumbic, M. Lipshteyn, Chordal probe graphs, Discrete Applied Mathematics 143 (2004) 221–237. [12] M.C. Golumbic, F. Maffray, G. Morel, A characterization of chain probe graphs, Annals of Operations Research (2009), in press. [13] M.C. Golumbic, A.N. Trenk, Tolerance Graphs, Cambridge University Press, 2004 (Chapter 4). [14] M. Habib, E. Lebhar, C. Paul, A note on finding all homogeneous set sandwiches, Information Processing Letters 87 (3) (2003) 147–151. [15] J.L. Johnson, J.P. Spinrad, A polynomial time recognition algorithm for probe interval graphs, in: Proc. SODA’01 (12th Annual ACM–SIAM Symposium on Discrete Algorithms), 2001, pp. 477–486. [16] R.M. McConnell, J.P. Spinrad, Construction of probe interval models, in: Proc. SODA’02 (13th Annual ACM–SIAM Symposium on Discrete Algorithms), 2002, pp. 866–875. [17] F.R. McMorris, C. Wang, P. Zhang, On probe interval graphs, Discrete Applied Mathematics 88 (1998) 315–324. [18] P. Zhang, Probe interval graphs and their application to physical mapping of DNA, manuscript, 1994.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
On the bi-enhancement of chordal-bipartite probe graphs Elad Cohen a,∗ , Martin Charles Golumbic a , Marina Lipshteyn a , Michal Stern a,b a b
Caesarea Rothschild Institute, and Department of Computer Science, University of Haifa, Haifa, Israel Academic College of Tel-Aviv-Jaffa, Israel
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 17 March 2009 Received in revised form 11 November 2009 Accepted 7 December 2009 Available online 11 December 2009 Communicated by S.E. Hambrusch Keywords: Combinatorial problems Chordal bipartite graphs Probe graphs
Given a class C of graphs, a graph G = ( V , E ) is said to be a C -probe graph if there exists a stable (i.e., independent) set of vertices X ⊆ V and a set F of pairs of vertices of X such that the graph G = ( V , E ∪ F ) is in the class C . Recently, there has been increasing interest and research on a variety of C -probe graph classes, such as interval probe graphs, chordal probe graphs and chain probe graphs. In this paper we focus on chordal-bipartite probe graphs. We prove a structural result that if B is a bipartite graph with no chordless cycle of length strictly greater than 6, then B is chordal-bipartite probe if and only if a certain “enhanced” graph B ∗ is a chordal-bipartite graph. This theorem is analogous to a result on interval probe graphs in Zhang (1994) [18] and to one on chordal probe graphs in Golumbic and Lipshteyn (2004) [11]. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Let G = ( V , E ) be a finite, undirected, simple graph. We say that vertex u “sees” vertex v if (u , v ) ∈ E. A set X ⊆ V is a stable set in G if for all u , v ∈ X , (u , v ) ∈ / E, i.e., no vertex in X sees another vertex in X . A graph G is a bipartite graph if its vertices can be partitioned into two disjoint stable sets V = X ∪ Y . We will refer to this as the “black/white” bipartition of the vertices. A sequence ( v 1 , . . . , v n ) of distinct vertices is a path in G if ( v i , v i +1 ) ∈ E for 1 i n − 1. A closed path ( v 1 , . . . , v n , v 1 ) is called a cycle if in addition ( v n , v 1 ) ∈ E. A chord of a cycle ( v 1 , . . . , v n , v 1 ) is an edge between two vertices of the cycle that is not an edge of the cycle. A chordless cycle C n is a cycle which contains no chords and has n vertices and n edges. A graph G is a chordal graph if it contains no induced chordless cycle C n , for n 4. A graph G is a chordalbipartite graph if it is a bipartite graph and it contains no induced chordless cycle C n , for n 6, as introduced in [8]. (Note that just as a chordal graph may contain triangles, a chordal-bipartite graph may contain chordless 4-cycles.)
*
Corresponding author. E-mail address: [email protected] (E. Cohen).
0020-0190/$ – see front matter doi:10.1016/j.ipl.2009.12.003
© 2009 Elsevier B.V.
All rights reserved.
Let C be a graph class. A graph G = ( V , E ) is called a C -probe graph if its vertex set can be partitioned into two subsets, P (probes) and N (non-probes), where N is a stable set, and one can add to G a set of edges between pairs of non-probes such that the resulting graph is in C . More formally, G is C -probe if there exists a completion F ⊆ {(u , v ) | u , v ∈ N , u = v } such that G = ( V , E ∪ F ) is in C . Recognizing whether or not G is a C -probe graph is known as the non-partitioned probe problem. In the partitioned version of the probe problem, the partition of V into probes and non-probes is given and fixed as part of the input. The partitioned C -probe problem is a special case of the C -sandwich problem [3,9,14]. Interval probe graphs were first introduced by Zhang [18] and studied further in [10,15–17], see also [13]. Chordal probe graphs were investigated in [11] and a characterization and recognition algorithm was given in [2] for both the partitioned and non-partitioned versions. Recently, characterizations and recognitions have been given for chain probe graphs in [12], and for threshold probe graphs and trivially-perfect probe graphs, also known as quasi-threshold probe graphs, in [4]. Other probe classes are to be found in [5,6]. In this paper, we focus on the partitioned version of chordal-bipartite probe graphs. In particular, we prove a
194
E. Cohen et al. / Information Processing Letters 110 (2010) 193–197
structural result that if B is a bipartite graph with no chordless cycle of length strictly greater than 6, then B is chordal-bipartite probe if and only if a certain “enhanced” graph B ∗ is a chordal-bipartite graph. This theorem is analogous to a result on interval probe graphs in [18] and to one on chordal probe graphs in [11]. We believe it may also shed light on the more general case of the partitioned version, where B may have chordless cycles of length larger than 6. We conclude the paper with open questions. All standard definitions can be found in [7]. 2. Motivation: chordal probe graphs Let G = ( V , E ) be a graph whose vertices have been partitioned into a set P of probes and a stable set N of non-probes. The following was proved in [11]: Lemma 2.1. (See [11].) If G is a chordal probe graph with respect to the partition P ∪ N, then probes and non-probes must alternate on every chordless cycle. In the specific case of a chordless 4-cycle with edges ab, bc , cd, da, this means that either {a, c } are probes and {b, d} are non-probes, or vice versa. Moreover, suppose that {a, c } are probes, then any possible chordal probe completion of G would be forced to contain the addition of an edge bd, which Zhang [18] called an enhanced edge. The enhanced graph G ∗ = ( P ∪ N , E ∗ ) is defined to be the graph obtained from G by adding all enhanced edges from all chordless 4-cycles. Theorem 2.2. (See [11].) Let G be a graph containing no induced chordless cycles C k for k > 4. If G has a probe/non-probe partition in which probes and non-probes alternate on every chordless 4-cycle, then the enhanced graph G ∗ is a chordal completion of G. Lemma 2.1 together with Theorem 2.2 prove the next corollary.
of the vertices and the “probe/non-probe” bipartition of the vertices are generally different! Moreover, if B is connected, then the completion edges between non-probes will have one endpoint white and the other black, in order to maintain the bipartite property. We begin by stating some necessary properties due to Berry, et al. [1]. The reader may wish to reconstruct a proof. Lemma 3.1. If B is a chordal-bipartite probe graph with respect to the partition P ∪ N, then on every chordless cycle of length 6 in B the following must hold: (1) every probe sees at most one other probe, (2) there is at least one edge of the cycle whose endpoints are both probes. Lemma 3.1 implies, in particular, that (1) on a chordless cycle there is no consecutive triple of probes, and that (2) there must be at least two pairs of consecutive probes, due to the parity of a bipartite graph. Moreover, Remark 3.2. In the specific case of a chordless 6-cycle C 6 , there are exactly 2 non-probes, one white and one black, opposite each other, and any possible chordal-bipartite probe completion of B must contain the added edge joining them. We will call this forced edge a bi-enhanced edge. Thus, with respect to a given probe/non-probe partition, we define the bi-enhanced graph B ∗ to be the graph obtained from B by adding all bi-enhanced edges from all chordless 6-cycles. We are now ready to state our main result. Theorem 3.3. Let B be a bipartite graph that contains no chordless cycle C k for k > 6. If B has a probe/non-probe partition in which probes and non-probes satisfy the property in Remark 3.2 on every chordless 6-cycle, then B ∗ is a chordal-bipartite completion of B.
Corollary 2.3. If G is a C k -free graph for k > 4, then G is chordal probe if and only if G ∗ is chordal.
Similar to the case of chordal probe graphs, Lemma 3.1 together with Theorem 3.3 immediately prove the next corollary.
This also gave an alternate proof of a result of Zhang [18].
Corollary 3.4. If B is a C k -free bipartite graph for k > 6, then B is chordal-bipartite probe if and only if B ∗ is chordal-bipartite.
Corollary 2.4. If G is an interval probe graph, then G ∗ is chordal.
See Fig. 1 for an example of a completed chordalbipartite probe graph with all its bi-enhanced edges.
3. Chordal-bipartite probe graphs We now turn our attention to the probe problem for chordal-bipartite graphs. Our goal will be to obtain a result in the same spirit as Corollary 2.3 in the case of a bipartite graph that has no chordless cycles of size strictly greater than 6. Let B = ( P ∪ N , E ) be a bipartite graph whose vertex set is partitioned into P (probes) and N (non-probes), where N is always assumed to be a stable set. Note that in the case of a bipartite graph B, the “black/white” bipartition
4. Proof of Theorem 3.3 This section is devoted to the proof of Theorem 3.3. Proof. Clearly, B ∗ is a bipartite graph. We will prove that B ∗ has no chordless cycle of size greater than 4. Suppose B ∗ has a chordless cycle of size greater than 4 and let h be the minimal number of bi-enhanced edges in such a cycle. Let C = (x1 , . . . , xm , x1 ) be a chordless cycle in B ∗ with h bi-enhanced edges.
E. Cohen et al. / Information Processing Letters 110 (2010) 193–197
195
on C and ai has exactly one neighbor on C , namely x1 , then b i is adjacent to all vertices of opposite color on C .
Fig. 1. Example of a completion of a chordal bipartite probe graph. Black and white vertices correspond to the bipartition sets.
If h = 0, then C is also a chordless cycle in B, and therefore C is of size 6. By Remark 3.2, C must have a chord which is a bi-enhanced edge. Contradiction! Thus, we may assume that h > 0. Without loss of generality, we may assume that (x1 , x2 ) is a bi-enhanced edge in C , and that (x1 , x2 ) was added in the construction of B ∗ as the chord of a chordless cycle C = (x1 , a1 , b1 , x2 , b2 , a2 , x1 ) in B of size 6. Claim 4.1. Suppose ai ∈ / C , for i = 1 or i = 2, then either ai has exactly one neighbor on C , namely x1 , or ai is also adjacent to xm−1 (see Fig. 2). Respectively, suppose b i ∈ / C , then either bi has exactly one neighbor on C , namely x2 , or b i is also adjacent to x4 .
Proof. We prove the first case only, since the second case is similar. Suppose there exists xl ∈ C that is of opposite color, but is not adjacent to ai , such that l is maximal. By Claim 4.1, ai is adjacent to xm−1 and thus l < m − 1. Suppose none of xl−1 , . . . , x3 is adjacent to ai . Let C = (ai , xl+2 , . . . , x3 , x2 , bi , ai ). Since (ai , xl+2 ) ∈ E ( B ), the cycle C is chordless cycle of size greater than 4 with at most h − 1 bi-enhanced edges. Contradiction! Otherwise, let xk ∈ C be a neighbor of ai , such that k is maximal and k < l. Since (ai , xk ) ∈ E ( B ), the cycle (ai , xl+2 , . . . , xk , ai ) is a chordless cycle of size greater than 4 with at most h − 1 bi-enhanced edges. Contradiction! 2 Claim 4.3. At most one of a1 , a2 , b1 , b2 is on C . Proof. Without loss of generality, suppose that a1 ∈ C . It immediately follows that a1 = xm , since C is chordless / C , since otherand (x1 , a1 ) ∈ E ( B ). Then obviously a2 ∈ / C , since otherwise wise a2 = a1 = xm . In addition, b1 ∈ (a1 , b1 ) is a chord in C . Suppose that b2 ∈ C , meaning that b2 = x3 . The cycle (x1 , a2 , b2 = x3 , x4 , . . . , xm = a1 , x1 ) is not chordless in B ∗ , since otherwise it has at most h − 1 bi-enhanced edges, contradicting the minimality of C . Therefore, a2 has more than one neighbor on C . Thus, by Claim 4.1, a2 is adjacent to xm−1 . Hence, (xm−1 , a2 , b2 = x3 , x2 , b1 , a1 = xm , xm−1 ) is a chordless cycle of size greater than 4 in B ∗ with at most h − 1 bi-enhanced edges, contradicting the minimality of C . 2
Proof. Suppose ai has more than one neighbor on C . Let x j ∈ C be the neighbor of ai , such that j is maximal. Then C = (ai , x1 , xm , . . . , x j , ai ) is a chordless cycle with at most h − 1 bi-enhanced edges, since (ai , x j ) ∈ E ( B ). By minimality of h in C , C must be of size 4 ( j = m − 1) and xm−1 is adjacent to ai . By symmetry, either b i has exactly one neighbor on C or b i is adjacent to x4 . 2
Definition 4.4 (bypass). Let (x1 , x2 ) be a bi-enhanced edge on C in B ∗ . Let [x1 , a, b, x2 ] be a chordless path in B, where a, b are probes. The pair a, b is called a bypass of / C and the edge (x1 , x2 ) with respect to C in B ∗ if a, b ∈ each of a, b has exactly one neighbor on C , namely, x1 and x2 , respectively.
Claim 4.2. Suppose V (C ) ∩ V (C ) = {x1 , x2 }. If ai has more than one neighbor on C and b i has exactly one neighbor on C , namely x2 , then ai is adjacent to all vertices of opposite color on C (see Fig. 2). Respectively, if b i has more than one neighbor
Proof. We prove that the bi-enhanced edge (x1 , x2 ) has a bypass. By Claim 4.3, at most one of a1 , a2 , b1 , b2 is on C . Therefore, there are two cases:
Claim 4.5. Every bi-enhanced edge on C in B ∗ has a bypass.
Case (i): Exactly one of a1 , b1 , a2 , b2 is on C .
Fig. 2. Examples of graphs B ∗ : (a) example where h = 1, under the conditions described in Claim 4.1; (b) example where h > 1 and some of the edges (xi , xi +1 ) are bi-enhanced, under the conditions described in Claim 4.2.
196
E. Cohen et al. / Information Processing Letters 110 (2010) 193–197
Fig. 3. Example of a graph B ∗ under the conditions described in Case (i).
Without loss of generality, suppose that a2 ∈ C , meaning that a2 = xm . Suppose a1 has more than one neighbor on C . Then, by Claim 4.1, a1 is adjacent to xm−1 . Consider the cycle C = (a2 = xm , xm−1 , a1 , b1 , x2 , b2 , a2 ). Due to the bipartition (xm−1 , b2 ), (xm−1 , b1 ) ∈ / E ( B ∗ ). Since a1 , a2 are probes, (xm−1 , a1 ), (xm−1 , a2 ) ∈ E ( B ). Then C is a chordless cycle of size 6 in B with no bi-enhanced edges, a contradiction. Therefore, a1 has exactly one neighbor on C . Suppose b1 has more than one neighbor on C . Let xi be a neighbor of b1 , such that i is maximal. Then, the cycle (a2 = xm , . . . , xi , b1 , a1 , x1 , a2 ) is chordless cycle of size greater than 4 in B ∗ with at most h − 1 bi-enhanced edges, contradicting the minimality of C . Therefore, b1 has only one neighbor on C . Thus, a1 , b1 is a bypass. Case (ii): None of a1 , b1 , a2 , b2 is on C . If a1 and b1 , and/or a2 and b2 have only one neighbor on C , then a1 , b1 and/or a2 , b2 are bypasses. Assume this is not the case. Suppose a1 and a2 both have more than one neighbor on C . Then, by Claim 4.1, a1 and a2 are adjacent / E ( B ∗ ), the cycle to xm−1 . Since (xm−1 , b1 ), (xm−1 , b2 ) ∈ (a1 , xm−1 , a2 , b2 , x2 , b1 , a1 ) is a chordless cycle of size 6 in B with no bi-enhanced edges. Contradiction! Thus, we may assume that a1 has only one neighbor on C . Similarly, at most one of b1 and b2 have more than one neighbor on C . Suppose b1 has more than one neighbor on C . Then, by Claim 4.1, (b1 , x4 ) ∈ E ( B ∗ ). Since, in this case, b2 has only one neighbor on C , then by Claim 4.2, (a2 , x3 ) ∈ E ( B ∗ ). Since (x3 , b1 ) ∈ / E ( B ∗ ) due to the bipartition, the cycle (a2 , x1 , a1 , b1 , x4 , x3 , a2 ) is a chordless cycle of size 6 in B ∗ with at most h − 1 bi-enhanced edges. Contradiction! Thus, b1 has only one neighbor on C , and a1 , b1 is a bypass. 2 Claim 4.6. A vertex cannot be in bypasses of different bienhanced edges in B ∗ . Proof. Suppose there exists a vertex u that participates in two bypasses a, b and a , b of two bi-enhanced edges (xi , xi +1 ) and (x j , x j +1 ) in C , respectively, such that i < j. Since u has only one neighbor in C , the two edges are consecutive, meaning that xi +1 = x j and u = b = a . Therefore, (xi , xi −1 , . . . , x j +1 , b , b, a, xi ) is a chordless cycle with
at most h − 2 bi-enhanced edges, contradicting the minimality of C . 2 We are now ready to complete the proof of the theorem. Let C be a cycle in B obtained from C , by substituting each bi-enhanced edge with the three edges of a bypass of that edge, due to Claim 4.6. Suppose C is a chordless cycle in B. Since h > 0 and C is a cycle in B ∗ with at least 6 vertices, the cycle C has at least 8 vertices. This is a contradiction, since B has no chordless cycle of size greater than 6. Therefore, C has a chord in B. Since C is chordless, and since each vertex in a bypass has only one neighbor on C , all such chords of C connect only bypass vertices of different bi-enhanced edges. Therefore, the theorem holds for h = 1 and we may now assume that h 2. Definition 4.7 (bridge). Let (xi , xi +1 ) and (x j , x j +1 ) be bienhanced edges on C in B ∗ with bypasses a, b and a , b , respectively, such that i < j. An edge between the two bypasses a, b and a , b is called a bridge. Notice that all the chords of C are bridges, and there may be at most two bridges between any pair a, b and a , b , since the graph is bipartite. There are two cases: Case (i): (a, b ) or (a , b) or both are bridges. Suppose (a, b ) is a bridge. Since C is a chordless cycle of size at least 6, at least one of the cycles (xi , a, b , a , x j +1 , . . . , xi ), (xi +1 , b, a, b , x j , . . . , xi +1 ) is a chordless cycle of size greater than 4 with at most h − 2 bi-enhanced edges, a contradiction (see Fig. 3). The case where (a , b) is a bridge is similar. Case (ii): (a, a ) or (b, b ) or both are bridges. Suppose (a, a ) is a bridge and (b, b ) is not a bridge. Since C is a chordless cycle of size at least 6, at least one of the cycles (xi , a, a , x j +1 , . . . , xi ), (xi +1 , . . . , x j , b , a , a, b, xi +1 ) is a chordless cycle of size greater than 4 with at most h − 2 bi-enhanced edges, a contradiction. The case where only (b, b ) is a bridge is similar. Suppose both (a, a ) and (b, b ) are bridges. Since C is a chordless cycle of size at least 6, at least one of the cycles (xi , a, a , x j +1 , . . . , xi ), (xi +1 , . . . , x j , b , b, xi +1 ) is a chordless cycle of size greater than 4 with at most h − 2 bienhanced edges, a contradiction.
E. Cohen et al. / Information Processing Letters 110 (2010) 193–197
Thus, there is no chordless cycle C of size greater than 4 in B ∗ . Hence, B ∗ is a chordal-bipartite graph. 2 5. Open questions In this paper, we prove that the properties of Lemma 3.1 characterize chordal-bipartite probe graphs in the case where the given graph contains no chordless cycles of size greater than 6. Berry et al. [1] have asked, what further conditions may be needed for these properties to characterize chordal-bipartite probe graphs in the general case? On the algorithmic side, given a probe/non-probe partition of a bipartite graph B, how do we most efficiently find the bi-enhanced edges, that is, build B ∗ ? What is the complexity of recognition of chordal-bipartite probe graphs, even in the case where there are no chordless cycles of size greater than 6? References [1] A. Berry, E. Cohen, M.C. Golumbic, M. Lipshteyn, N. Pinet, A. Sigayret, M. Stern, Recognizing chordal-bipartite probe graphs, France–Israel binational science collaboration project, unpublished technical report, 2007. [2] A. Berry, M.C. Golumbic, M. Lipshteyn, Recognizing chordal probe graphs and cycle-bicolorable graphs, SIAM Journal on Discrete Mathematics 21 (2007) 573–591. [3] C. Bornstein, C.M.H. de Figueiredo, V.G.P. de Sá, The pair completion algorithm for the homogeneous set sandwich problem, Information Processing Letters 98 (3) (2006) 87–91. [4] D. Bayera, V.B. Leb, H.N. de Ridder, Probe threshold and probe trivially perfect graphs, Theoretical Computer Science 410 (47–49) (2009) 4812–4822.
197
[5] D.B. Chandler, M.S. Chang, T. Kloks, S.L. Peng, Probe Graphs, preliminary version, 2009. [6] M.S. Chang, T. Kloks, D. Kratsch, J. Liu, S.-L. Peng, On the recognition of probe graphs of some self-complementary classes of perfect graphs, LNCS 3595 (2005) 808–817. [7] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980; M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, second edition, Annals of Discrete Mathematics, vol. 57, Elsevier, Amsterdam, 2004. [8] M.C. Golumbic, C.F. Goss, Perfect elimination and chordal-bipartite graphs, Journal of Graph Theory 2 (1978) 155–163. [9] M.C. Golumbic, H. Kaplan, R. Shamir, Graph sandwich problems, Journal of Algorithms 19 (1995) 449–473. [10] M.C. Golumbic, M. Lipshteyn, On the hierarchy of interval, probe and tolerance graphs, in: Proc. 32th Southeastern Conf. on Combinatorics, Graph Theory and Computing, in: Congressus Numerantium, vol. 153, Utilitas Math., Winnipeg, Canada, 2001, pp. 97–106. [11] M.C. Golumbic, M. Lipshteyn, Chordal probe graphs, Discrete Applied Mathematics 143 (2004) 221–237. [12] M.C. Golumbic, F. Maffray, G. Morel, A characterization of chain probe graphs, Annals of Operations Research (2009), in press. [13] M.C. Golumbic, A.N. Trenk, Tolerance Graphs, Cambridge University Press, 2004 (Chapter 4). [14] M. Habib, E. Lebhar, C. Paul, A note on finding all homogeneous set sandwiches, Information Processing Letters 87 (3) (2003) 147–151. [15] J.L. Johnson, J.P. Spinrad, A polynomial time recognition algorithm for probe interval graphs, in: Proc. SODA’01 (12th Annual ACM–SIAM Symposium on Discrete Algorithms), 2001, pp. 477–486. [16] R.M. McConnell, J.P. Spinrad, Construction of probe interval models, in: Proc. SODA’02 (13th Annual ACM–SIAM Symposium on Discrete Algorithms), 2002, pp. 866–875. [17] F.R. McMorris, C. Wang, P. Zhang, On probe interval graphs, Discrete Applied Mathematics 88 (1998) 315–324. [18] P. Zhang, Probe interval graphs and their application to physical mapping of DNA, manuscript, 1994.