- Email: [email protected]
Probe interval bigraphs
Electronic Notes in Discrete Mathematics 19 (2005) 195–201 www.elsevier.com/locate/endm
Probe interval bigraphs Gerard Jennhwa Chang 1,2 , Ton Kloks, Department of Mathematics National Taiwan University Taipei 10617, Taiwan
Sheng–Lung Peng 3 Department of Computer Science and Information Engineering National Dung Hua University Hualien 974, Taiwan
Abstract Given a class of graphs G, a graph G is a probe graph of G if its vertices can be partitioned into two sets P (the probes) and N (non–probes), where N is an independent set, such that G can be embedded into a graph of G by adding edges between certain vertices of N. We show that the recognition problem of probe interval bigraphs, i.e., probe graphs of the class of interval bigraphs, is in P. Keywords: recognition algorithm, probe interval bigraphs
1 Supported in part by the National Science Council under grant NSC93-2115-M002-003. Member of the Mathematics Division, National Center for Theoretical Sciences at Taipei. 2 Email: [email protected]. 3 Email: [email protected] (corresponding author).
1571-0653/2005 Published by Elsevier B.V. doi:10.1016/j.endm.2005.05.027
196
1
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
Introduction
Probe interval graphs were introduced in [9] to model certain problems in physical mapping of DNA. Chapter 4 in the recent book by Golumbic and Trenk [4] is dedicated to probe interval graphs. The application in molecular biology is the problem of reconstructing the arrangement of fragments of DNA taken from multiple copies of the same genome. The results of laboratory tests tell us which pairs of fragments occupy intersecting intervals of the genome. The genetic information is physically organized in a linear arrangement, and, when full information is available, an arrangement of intervals can be created in linear time. More recently, a variant that makes more efficient use of laboratory resources has been studied. A subset of the fragments is designated as probes, and for each probe one may test all non–probe fragments for intersection with the probe. In graph–theoretic terms, for the partitioned case, the input to the problem is a graph G and a subset of probe vertices. The other vertices, the non–probes, form an independent set in G. The objective is to add edges between certain non–probe vertices such that the graph becomes an interval graph. For the unpartitioned case, the input to the problem is just a graph G. The objective is to determine whether G is a probe interval graph or not. Note that in the unpartitioned case the probe set is unknown. Recently it was established that the class of unpartitioned probe interval graphs permits a polynomial time recognition algorithm [3]. It is mainly of theoretical interest to study other (unpartitioned) probe graph classes although also probe chordal graphs have immediate applications, e.g., in the reconstruction of phylogenies. From a theoretical point of view other probe graph classes are of great interest. First of all, it establishes boundaries on the robustness of recognizing the graph class with respect to irresolute inputs. The investigation of probe graph classes was contemplated initially for this objective. Also, the research into probe graph classes brings to light many interesting, sometimes unforeseen properties of the graph class in question. For example, it turns out that probe interval graphs are weakly chordal, hence perfect. The class of AT–free graphs, that properly contains the class of interval graph, contains imperfect graphs such as the C5. However, the independence number remains polynomial time computable.4 We suspect that this remains true for probe AT–free graphs. Also, some techniques seem 4
In contrast, computing the clique number remains an NP-complete problem.
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
197
apt for solving certain domination problems for these graphs. In this paper we deal with the class of probe interval bigraphs.
2
Preliminaries
For a graph G = (V, E) and a subset S ⊆ V we write G[S] for the subgraph of G induced by S, i.e., the subgraph with vertex set S and edges those elements of E with both end–vertices in S. For a subset W ⊆ V we write G − W for the subgraph induced by V − W. For a vertex x we write G − x rather than G − {x}. Let N(x) = {y | (x, y) ∈ E} and N[x] = N(x) ∪ {x}. Sometimes we write x ∼ y if x = y and x and y are adjacent in G. For an edge e = (x, y) we write N(e) = N(x) + N(y) and call this the neighborhood of e. A graph is bipartite if its set of vertices can be partitioned into two color classes, say X and Y, such that there are no edges that are 2–subsets contained within the same color class. We represent a bipartite graph as B = (X, Y, E). A bipartite graph B = (X, Y, E) is complete bipartite if every vertex of X is adjacent to every vertex of Y. If B is complete bipartite, we also write B = (X, Y). A biclique in a bipartite graph B = (X, Y, E) is a complete bipartite subgraph. For other graph classes not defined in this paper here we refer to [2]. Definition 2.1 A bipartite graph B = (X, Y, E) is chordal bipartite if it does not have an induced cycle of length more than 4. A graph is weakly chordal if it doesn’t contain any chordless cycle of length more than 4 or the complement of such a cycle. Hence chordal bipartite graphs are exactly the bipartite weakly chordal graphs.5 Many other characterizations appeared in terms of edge separators, matrices, and elimination orderings. See [2] for a compendium. Definition 2.2 A probe chordal bipartite graph is a bipartite graph B for which there exists a partition of the vertices into probes P and non–probes N such that N is an independent set and B can be made chordal bipartite by adding some edges between vertices of N. Remark 2.3 Notice that a probe chordal bipartite graph can have arbitrarily large (even) cycles. At this point, we remind the reader that, for any fixed constant k, a chordless cycle of length at least k can be found in polynomial time [8]. 5 It yields to argument that they should be called this way, since chordal bipartite graphs are not chordal.
198
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
Definition 2.4 An embedding of B = (X, Y, E) is a chordal bipartite graph B = (X, Y, E ) together with a partition of X + Y into probes and non–probes such that all edges of E − E are edges between non–probes. Remark 2.5 Notice that B has the same color classes as B. This is no restriction, since, as long as B is connected, the color classes are uniquely defined. Definition 2.6 An edge of B is a probe edge if its two end–vertices are probes. We start with some observation on the embedding of chordless cycles of length more than 4. If Ω is such a chordless cycle in B of length more than 4 then: (i) A probe vertex has no other neighbors in B than its neighbors in B. The neighbors of a probe edge are connected in B. (ii) Also, Ω has no three consecutive probes, otherwise there is an induced cycle of length more than 4 in B. (iii) There are at least two probe edges, since, if all edges have one probe and one non–probe end–vertex, then all non–probes are in one color class. But, in B, there are no edges between vertices of the same color class. The number of probe edges in Ω is even since otherwise Ω would be an odd cycle contradicting that B is bipartite. (iv) Every chord in the embedding, has an odd number of probe edges on both sides, otherwise B has an odd cycle.
3
Interval bigraphs
Interval bigraphs were introduced in [5] as those bipartite graphs B = (X, Y, E) for which there is a collection of intervals on the real line and a 1–1 mapping from the vertices of B to these intervals, such that two vertices x ∈ X and y ∈ Y are adjacent if and only if their corresponding intervals intersect. Validating if a bipartite graph is in this class can be done in polynomial time [7].6 The class contains all bipartite permutation graphs. Recall that an interval graph G is the intersection graph of a family of intervals of the real line. Furthermore, the maximal cliques of G can be linearly ordered such that for every vertex x, the maximal cliques containing x occur consecutively. In the same phraseology, we have the following result. 6
The time bound, given in this paper, is O(nm6 (n + m) log n).
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
199
Theorem 3.1 Let B be an interval bigraph. Then there is a linear arrangement B = [B1, . . . , Bt] of some of its maximal bicliques such that for every edge e of B, there is at least one Bi containing e and for every vertex v, the bicliques containing v occur consecutively in B. Proof. Consider an interval model I of B. Let G(I) be the interval graph corresponding to I. Let C = [C1, . . . , Ct] be the linear ordered maximal cliques of G(I). In B, each Ci is a biclique and is denoted as Bi. Then [B1, . . . , Bt] is a consecutive arrangement of some of B’s maximal bicliques. Conversely, if there is a linear arrangement B = [B1, . . . , Bt] for B such that each edge is contained in a Bi and each vertex v occurs consecutively in bicliques of B. Assume that v occurs in [Bi, . . . , Bj] for some i ≤ j. Then let Iv = [i, j] be the interval corresponding to v for each v. It is not hard to check that I = {Iv | v ∈ V(B)} is an interval model of B. This completes the proof.2 Definition 3.2 An ATE is a set of three edges such that for any two of them there is a path from the vertex set of one to the vertex set of the other that avoids the neighborhood of the third. It is well–known (see, e.g., [7]) that interval bigraphs are ATE–free, i.e., they do not have an ATE. A similar characterization appeared in [5] using bi−AT’s. A bi−AT is a triple of vertices such that for every two of them there is a connecting path such that the third vertex has distance more than 2 from this path. Every ATE–free graph is bi−AT–free [7]. In [7] a conjecture appeared saying that a bipartite graph is an interval bigraph if and only if it is ATE–free and insect–free.7 Theorem 3.3 Let B be a probe interval bigraph. Every chordless cycle of length more than 4 has exactly two probe edges, and they are at even distance. Proof. Every chordless cycle of (even) length at least 6 has a number of chords in B. If there are more than two probe edges, there would be an ATE in B, since the two neighbors of every probe edge are connected non–probes in B. The two probe edges are connected by two alternating probe–non–probe paths. 2 Definition 3.4 Two edges e = (x, y) and f = (a, b) are distinct if {a, b} ∩ {x, y} = ∅. They are disjoint if they are distinct and a, b ∈ N(e). 7
An insect is a graph on 12 vertices consisting of a K3,3 and a matching from these vertices to the remaining vertices called the feet. Additional edges between the feet are allowed.
200
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
Theorem 3.5 Let B be a biconnected probe interval bigraph. Then for every edge e = (x, y) there are at most two components C1 and C2 of B − {x, y} that contain chordless paths of length at least 3 that make up chordless cycles with e of length at least 6. Proof. Assume there are three components C1, C2, C3 of B − {x, y}, each containing a chordless path of length at least 3 with the end–vertices connected to x and y respectively. Each of the 3 cycles contains a probe edge distinct from (x, y). Since (x, y) is an edge in B, at least one of x and y is in P. If x ∈ P then the three probe edges are disjoint from x. Hence we obtain an ATE. 2
4
Recognition of probe interval bigraphs
Throughout this section we assume that B is connected. Let e = (x, y) be a probe edge in some embedding B. Consider the following procedure: For every chordless cycle Ω of length at least 6 that contains e, connect the two neighbors of x and y in Ω. Let Be be the graph obtained in this way. Theorem 4.1 Let e = (x, y) be a probe edge in some embedding B. Let C be a component of B − {x, y}. Let NC(e) ⊆ N(e) be the set of vertices of N(e) that have a neighbor in C. Then NC(e) is complete bipartite in Be. Proof. Assume NC(e) has two non–adjacent vertices a and b in opposite color classes. Let a ∼ x and b ∼ y. Since a, b ∈ NC(e), a and b have neighbors a and b in C. Since C is connected, there is a path connecting a and b with vertices in C. Thus we can derive a chordless cycle Ω of length at least 6 that contains e with neighbors a and b on this cycle. But then, 2 according to the construction of Be, a and b are connected. Clearly, also NC(e) + x + y is complete bipartite. We write NC[e] = NC(e) + x + y and call this the neighborhood biclique of e with respect to C. Theorem 4.2 There exists a polynomial time algorithm to recognize probe interval bigraphs. Proof. In this extended abstract we only consider the case where B is biconnected. Further details are included in the full paper. Let e = (x, y) and f = (a, b) be two disjoint edges. Let Ce(f) be the component of B−{x, y} that contains the edge f and let Cf(e) be the component of B − {a, b} that contains e. The interval I(e, f) between e and f is defined as
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
201
(Ce(f) ∩ Cf(e)) + N[e] + N[f]. An interval is valid if there is an embedding of the induced subgraph such that there is a linear arrangement of the bicliques with e contained in the first and f in the last biclique. Sort the intervals according to their number of vertices. If the interval is a chordless (even) cycle then it is valid if and only if e and f are at even distance. Consider an interval I(e, f). Try all edges s = (c, d) in I(e, f) disjoint from e and f such that N(s) separates e and f. There can be at most two components in B − {c, d} that contain probe edges. One component contains e and the other contains f. The other components must form complete bipartite subgraphs including s in the embedding. If this is the case, and if I(e, s) and I(s, f) are both valid, then I(e, f) is valid as well. Further details can be found in the full paper. 2
References [1] Berry, A., M. C. Golumbic, and M. Lipschteyn, Two tricks to triangulate chordal probe graphs in polynomial time, Proceedings of the 15th ACM–SIAM Symposium on Discrete Algorithms, 2004, pp. 962–969. [2] Br¨ andstadt, A., V. Le, and J. P. Spinrad, Graph classes – A survey, SIAM Monographs on Discrete Mathematics and Applications, Philadelphia, 1999. [3] Chang, G. J., A. J. J. Kloks, J. Liu, and S. L. Peng, The PIGs full monty–A floor show of minimal separators. To appear at STACS 2005, LNCS 3404. [4] Golumbic, M. C. and A. N. Trenk, Tolerance Graphs, Cambridge studies in advanced mathematics, Cambridge University Press, 2004. [5] Harary, F., J. A. Kabell, and F. R. McMorris, Interval bigraphs, Comment. Math. Univ. Carolinae 23 (1982), pp. 739–745. [6] Kloks, T., Treewidth–Computations and Approximations, Springer–Verlag, LNCS 842, 1994. [7] M¨ uller, H., Recognizing interval digraphs and interval bigraphs in polynomial time, Discrete Applied Mathematics 78 (1997), pp. 189–205. Erratum, http://www.comp.leeds.ac.uk/hm/publ.html. [8] Spinrad, J. P., Finding large holes, Information Processing Letters 39 (1991), pp. 227–229. [9] Zhang, P., Probe interval graph and its application to physical mapping of DNA. Manuscript 1994.
Probe interval bigraphs Gerard Jennhwa Chang 1,2 , Ton Kloks, Department of Mathematics National Taiwan University Taipei 10617, Taiwan
Sheng–Lung Peng 3 Department of Computer Science and Information Engineering National Dung Hua University Hualien 974, Taiwan
Abstract Given a class of graphs G, a graph G is a probe graph of G if its vertices can be partitioned into two sets P (the probes) and N (non–probes), where N is an independent set, such that G can be embedded into a graph of G by adding edges between certain vertices of N. We show that the recognition problem of probe interval bigraphs, i.e., probe graphs of the class of interval bigraphs, is in P. Keywords: recognition algorithm, probe interval bigraphs
1 Supported in part by the National Science Council under grant NSC93-2115-M002-003. Member of the Mathematics Division, National Center for Theoretical Sciences at Taipei. 2 Email: [email protected]. 3 Email: [email protected] (corresponding author).
1571-0653/2005 Published by Elsevier B.V. doi:10.1016/j.endm.2005.05.027
196
1
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
Introduction
Probe interval graphs were introduced in [9] to model certain problems in physical mapping of DNA. Chapter 4 in the recent book by Golumbic and Trenk [4] is dedicated to probe interval graphs. The application in molecular biology is the problem of reconstructing the arrangement of fragments of DNA taken from multiple copies of the same genome. The results of laboratory tests tell us which pairs of fragments occupy intersecting intervals of the genome. The genetic information is physically organized in a linear arrangement, and, when full information is available, an arrangement of intervals can be created in linear time. More recently, a variant that makes more efficient use of laboratory resources has been studied. A subset of the fragments is designated as probes, and for each probe one may test all non–probe fragments for intersection with the probe. In graph–theoretic terms, for the partitioned case, the input to the problem is a graph G and a subset of probe vertices. The other vertices, the non–probes, form an independent set in G. The objective is to add edges between certain non–probe vertices such that the graph becomes an interval graph. For the unpartitioned case, the input to the problem is just a graph G. The objective is to determine whether G is a probe interval graph or not. Note that in the unpartitioned case the probe set is unknown. Recently it was established that the class of unpartitioned probe interval graphs permits a polynomial time recognition algorithm [3]. It is mainly of theoretical interest to study other (unpartitioned) probe graph classes although also probe chordal graphs have immediate applications, e.g., in the reconstruction of phylogenies. From a theoretical point of view other probe graph classes are of great interest. First of all, it establishes boundaries on the robustness of recognizing the graph class with respect to irresolute inputs. The investigation of probe graph classes was contemplated initially for this objective. Also, the research into probe graph classes brings to light many interesting, sometimes unforeseen properties of the graph class in question. For example, it turns out that probe interval graphs are weakly chordal, hence perfect. The class of AT–free graphs, that properly contains the class of interval graph, contains imperfect graphs such as the C5. However, the independence number remains polynomial time computable.4 We suspect that this remains true for probe AT–free graphs. Also, some techniques seem 4
In contrast, computing the clique number remains an NP-complete problem.
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
197
apt for solving certain domination problems for these graphs. In this paper we deal with the class of probe interval bigraphs.
2
Preliminaries
For a graph G = (V, E) and a subset S ⊆ V we write G[S] for the subgraph of G induced by S, i.e., the subgraph with vertex set S and edges those elements of E with both end–vertices in S. For a subset W ⊆ V we write G − W for the subgraph induced by V − W. For a vertex x we write G − x rather than G − {x}. Let N(x) = {y | (x, y) ∈ E} and N[x] = N(x) ∪ {x}. Sometimes we write x ∼ y if x = y and x and y are adjacent in G. For an edge e = (x, y) we write N(e) = N(x) + N(y) and call this the neighborhood of e. A graph is bipartite if its set of vertices can be partitioned into two color classes, say X and Y, such that there are no edges that are 2–subsets contained within the same color class. We represent a bipartite graph as B = (X, Y, E). A bipartite graph B = (X, Y, E) is complete bipartite if every vertex of X is adjacent to every vertex of Y. If B is complete bipartite, we also write B = (X, Y). A biclique in a bipartite graph B = (X, Y, E) is a complete bipartite subgraph. For other graph classes not defined in this paper here we refer to [2]. Definition 2.1 A bipartite graph B = (X, Y, E) is chordal bipartite if it does not have an induced cycle of length more than 4. A graph is weakly chordal if it doesn’t contain any chordless cycle of length more than 4 or the complement of such a cycle. Hence chordal bipartite graphs are exactly the bipartite weakly chordal graphs.5 Many other characterizations appeared in terms of edge separators, matrices, and elimination orderings. See [2] for a compendium. Definition 2.2 A probe chordal bipartite graph is a bipartite graph B for which there exists a partition of the vertices into probes P and non–probes N such that N is an independent set and B can be made chordal bipartite by adding some edges between vertices of N. Remark 2.3 Notice that a probe chordal bipartite graph can have arbitrarily large (even) cycles. At this point, we remind the reader that, for any fixed constant k, a chordless cycle of length at least k can be found in polynomial time [8]. 5 It yields to argument that they should be called this way, since chordal bipartite graphs are not chordal.
198
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
Definition 2.4 An embedding of B = (X, Y, E) is a chordal bipartite graph B = (X, Y, E ) together with a partition of X + Y into probes and non–probes such that all edges of E − E are edges between non–probes. Remark 2.5 Notice that B has the same color classes as B. This is no restriction, since, as long as B is connected, the color classes are uniquely defined. Definition 2.6 An edge of B is a probe edge if its two end–vertices are probes. We start with some observation on the embedding of chordless cycles of length more than 4. If Ω is such a chordless cycle in B of length more than 4 then: (i) A probe vertex has no other neighbors in B than its neighbors in B. The neighbors of a probe edge are connected in B. (ii) Also, Ω has no three consecutive probes, otherwise there is an induced cycle of length more than 4 in B. (iii) There are at least two probe edges, since, if all edges have one probe and one non–probe end–vertex, then all non–probes are in one color class. But, in B, there are no edges between vertices of the same color class. The number of probe edges in Ω is even since otherwise Ω would be an odd cycle contradicting that B is bipartite. (iv) Every chord in the embedding, has an odd number of probe edges on both sides, otherwise B has an odd cycle.
3
Interval bigraphs
Interval bigraphs were introduced in [5] as those bipartite graphs B = (X, Y, E) for which there is a collection of intervals on the real line and a 1–1 mapping from the vertices of B to these intervals, such that two vertices x ∈ X and y ∈ Y are adjacent if and only if their corresponding intervals intersect. Validating if a bipartite graph is in this class can be done in polynomial time [7].6 The class contains all bipartite permutation graphs. Recall that an interval graph G is the intersection graph of a family of intervals of the real line. Furthermore, the maximal cliques of G can be linearly ordered such that for every vertex x, the maximal cliques containing x occur consecutively. In the same phraseology, we have the following result. 6
The time bound, given in this paper, is O(nm6 (n + m) log n).
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
199
Theorem 3.1 Let B be an interval bigraph. Then there is a linear arrangement B = [B1, . . . , Bt] of some of its maximal bicliques such that for every edge e of B, there is at least one Bi containing e and for every vertex v, the bicliques containing v occur consecutively in B. Proof. Consider an interval model I of B. Let G(I) be the interval graph corresponding to I. Let C = [C1, . . . , Ct] be the linear ordered maximal cliques of G(I). In B, each Ci is a biclique and is denoted as Bi. Then [B1, . . . , Bt] is a consecutive arrangement of some of B’s maximal bicliques. Conversely, if there is a linear arrangement B = [B1, . . . , Bt] for B such that each edge is contained in a Bi and each vertex v occurs consecutively in bicliques of B. Assume that v occurs in [Bi, . . . , Bj] for some i ≤ j. Then let Iv = [i, j] be the interval corresponding to v for each v. It is not hard to check that I = {Iv | v ∈ V(B)} is an interval model of B. This completes the proof.2 Definition 3.2 An ATE is a set of three edges such that for any two of them there is a path from the vertex set of one to the vertex set of the other that avoids the neighborhood of the third. It is well–known (see, e.g., [7]) that interval bigraphs are ATE–free, i.e., they do not have an ATE. A similar characterization appeared in [5] using bi−AT’s. A bi−AT is a triple of vertices such that for every two of them there is a connecting path such that the third vertex has distance more than 2 from this path. Every ATE–free graph is bi−AT–free [7]. In [7] a conjecture appeared saying that a bipartite graph is an interval bigraph if and only if it is ATE–free and insect–free.7 Theorem 3.3 Let B be a probe interval bigraph. Every chordless cycle of length more than 4 has exactly two probe edges, and they are at even distance. Proof. Every chordless cycle of (even) length at least 6 has a number of chords in B. If there are more than two probe edges, there would be an ATE in B, since the two neighbors of every probe edge are connected non–probes in B. The two probe edges are connected by two alternating probe–non–probe paths. 2 Definition 3.4 Two edges e = (x, y) and f = (a, b) are distinct if {a, b} ∩ {x, y} = ∅. They are disjoint if they are distinct and a, b ∈ N(e). 7
An insect is a graph on 12 vertices consisting of a K3,3 and a matching from these vertices to the remaining vertices called the feet. Additional edges between the feet are allowed.
200
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
Theorem 3.5 Let B be a biconnected probe interval bigraph. Then for every edge e = (x, y) there are at most two components C1 and C2 of B − {x, y} that contain chordless paths of length at least 3 that make up chordless cycles with e of length at least 6. Proof. Assume there are three components C1, C2, C3 of B − {x, y}, each containing a chordless path of length at least 3 with the end–vertices connected to x and y respectively. Each of the 3 cycles contains a probe edge distinct from (x, y). Since (x, y) is an edge in B, at least one of x and y is in P. If x ∈ P then the three probe edges are disjoint from x. Hence we obtain an ATE. 2
4
Recognition of probe interval bigraphs
Throughout this section we assume that B is connected. Let e = (x, y) be a probe edge in some embedding B. Consider the following procedure: For every chordless cycle Ω of length at least 6 that contains e, connect the two neighbors of x and y in Ω. Let Be be the graph obtained in this way. Theorem 4.1 Let e = (x, y) be a probe edge in some embedding B. Let C be a component of B − {x, y}. Let NC(e) ⊆ N(e) be the set of vertices of N(e) that have a neighbor in C. Then NC(e) is complete bipartite in Be. Proof. Assume NC(e) has two non–adjacent vertices a and b in opposite color classes. Let a ∼ x and b ∼ y. Since a, b ∈ NC(e), a and b have neighbors a and b in C. Since C is connected, there is a path connecting a and b with vertices in C. Thus we can derive a chordless cycle Ω of length at least 6 that contains e with neighbors a and b on this cycle. But then, 2 according to the construction of Be, a and b are connected. Clearly, also NC(e) + x + y is complete bipartite. We write NC[e] = NC(e) + x + y and call this the neighborhood biclique of e with respect to C. Theorem 4.2 There exists a polynomial time algorithm to recognize probe interval bigraphs. Proof. In this extended abstract we only consider the case where B is biconnected. Further details are included in the full paper. Let e = (x, y) and f = (a, b) be two disjoint edges. Let Ce(f) be the component of B−{x, y} that contains the edge f and let Cf(e) be the component of B − {a, b} that contains e. The interval I(e, f) between e and f is defined as
G.J. Chang et al. / Electronic Notes in Discrete Mathematics 19 (2005) 195–201
201
(Ce(f) ∩ Cf(e)) + N[e] + N[f]. An interval is valid if there is an embedding of the induced subgraph such that there is a linear arrangement of the bicliques with e contained in the first and f in the last biclique. Sort the intervals according to their number of vertices. If the interval is a chordless (even) cycle then it is valid if and only if e and f are at even distance. Consider an interval I(e, f). Try all edges s = (c, d) in I(e, f) disjoint from e and f such that N(s) separates e and f. There can be at most two components in B − {c, d} that contain probe edges. One component contains e and the other contains f. The other components must form complete bipartite subgraphs including s in the embedding. If this is the case, and if I(e, s) and I(s, f) are both valid, then I(e, f) is valid as well. Further details can be found in the full paper. 2
References [1] Berry, A., M. C. Golumbic, and M. Lipschteyn, Two tricks to triangulate chordal probe graphs in polynomial time, Proceedings of the 15th ACM–SIAM Symposium on Discrete Algorithms, 2004, pp. 962–969. [2] Br¨ andstadt, A., V. Le, and J. P. Spinrad, Graph classes – A survey, SIAM Monographs on Discrete Mathematics and Applications, Philadelphia, 1999. [3] Chang, G. J., A. J. J. Kloks, J. Liu, and S. L. Peng, The PIGs full monty–A floor show of minimal separators. To appear at STACS 2005, LNCS 3404. [4] Golumbic, M. C. and A. N. Trenk, Tolerance Graphs, Cambridge studies in advanced mathematics, Cambridge University Press, 2004. [5] Harary, F., J. A. Kabell, and F. R. McMorris, Interval bigraphs, Comment. Math. Univ. Carolinae 23 (1982), pp. 739–745. [6] Kloks, T., Treewidth–Computations and Approximations, Springer–Verlag, LNCS 842, 1994. [7] M¨ uller, H., Recognizing interval digraphs and interval bigraphs in polynomial time, Discrete Applied Mathematics 78 (1997), pp. 189–205. Erratum, http://www.comp.leeds.ac.uk/hm/publ.html. [8] Spinrad, J. P., Finding large holes, Information Processing Letters 39 (1991), pp. 227–229. [9] Zhang, P., Probe interval graph and its application to physical mapping of DNA. Manuscript 1994.