Norms and perfect graphs

The weak Berge hypothesis states that a graph is perfect if and only if its complement is perfect. Previous proofs of this hypothesis have used combinatorial or polyhedral methods.In this paper, the concept of norms related to graphs is used to provide an alternative proof for the weak Berge hypothesis.This is a written account of an invited lecture delivered by the second author on occasion of the 12. Symposium on Operations Research, Passau, 9.–11. 9. 1987.     

Progress on perfect graphs

 A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved efficiently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph. In the first part of the paper we survey the main aspects of perfect graphs and their relevance. In the second part we outline our recent proof of the Strong Perfect Graph Conjecture of Berge from 1961, the following: a graph is perfect if and only if it has no induced subgraph isomorphic to an odd cycle of length at least five, or the complement of such an odd cycle. Received: December 19, 2002 / Accepted: April 29, 2003 Published online: May 28, 2003 Key words. Berge graph – perfect graph – skew partition Mathematics Subject Classification (1991): 05C17     

On Perfect Matching Coverings and Even Subgraph Coverings

A perfect matching covering of a graph G is a set of perfect matchings of G such that every edge of G is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph admits a perfect matching covering of order at most 5 (we call such a collection of perfect matchings a Berge covering of G). A cubic graph G is called a Kotzig graph if G has a 3‐edge‐coloring such that each pair of colors forms a hamiltonian circuit (introduced by R. Häggkvist, K. Markström, J Combin Theory Ser B 96 (2006), 183–206). In this article, we prove that if there is a vertex w of a cubic graph G such that , the graph obtained from by suppressing all degree two vertices is a Kotzig graph, then G has a Berge covering. We also obtain some results concerning the so‐called 5‐even subgraph double cover conjecture.     

Chair-Free Berge Graphs Are Perfect

A graph G is called Berge if neither G nor its complement contains a chordless cycle with an odd number of nodes. The famous Berge’s Strong Perfect Graph Conjecture asserts that every Berge graph is perfect. A chair is a graph with nodes {a, b, c, d, e} and edges {ab, bc, cd}, eb. We prove that a Berge graph with no induced chair (chair-free) is perfect or, equivalently, that the Strong Perfect Graph Conjecture is true for chair-free graphs.     

A note on line coverings of graphs

Let minimum, maximum, refer to the number of elements in a set and let minimal, maximal refer to set inclusion. It is shown that a minimal cover of a graph is minimum if and only if it contains a maximum matching of that graph; a maximal matching of a graph is maximum if and only if it is contained in a minimum cover of that graph diminished by the set of its isolated points.     

Unicycle graphs and uniquely restricted maximum matchings

A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. G is a unicycle graph if it owns only one cycle. Golumbic, Hirst and Lewenstein observed that for a tree or a graph with only odd cycles the size of a maximum uniquely restricted matching is equal to the matching number of the graph. In this paper we characterize unicycle graphs enjoying this equality. Moreover, we describe unicycle graphs with only uniquely restricted maximum matchings. Using these findings, we show that unicycle graphs having only uniquely restricted maximum matchings can be recognized in polynomial time.     

3阶幂图的广义导出匹配可扩性

A graph G is induced matching extendable if every induced matching of G is included in a perfect matching of G. A graph G is generalized induced matching extendable if every induced matching of G is included in a maximum matching of G. A graph G is claw-free, if G dose not contain any induced subgraph isomorphic to K1,3. The k-th power of G, denoted by Gu, is the graph with vertex set V（G） in which two vertices are adjacent if and only if the distance between them is at most k in G. In this paper we show that, if the maximum matchings of G and G3 have the same cardinality, then G3 is generalized induced matching extendable. We also show that this result is best possible. As a result, we show that if G is a connected claw-flee graph, then G3 is generalized induced matching extendable.     

Perfect zero–one matrices

A zero–one matrix is called perfect if the polytope of the associated set packing problem has integral vertices only. By this definition, all totally unimodular zero–one matrices are perfect. In this paper we give a characterization of perfect zero–one matrices in terms offorbidden submatrices. Perfect zero–one matrices are closely related to perfect graphs and constitute a generalization of balanced matrices as introduced by C. Berge. Furthermore, the results obtained here bear on an unsolved problem in graph theory, the strong perfect graph conjecture, also due to C. Berge.     

Graphs isomorphic to their maximum matching graphs

The maximum matching graph M（G） of a graph G is a simple graph whose vertices are the maximum matchings of G and where two maximum matchings are adjacent in M（G） if they differ by exactly one edge. In this paper, we prove that if a graph is isomorphic to its maximum matching graph, then every block of the graph is an odd cycle.     

Bounding the Size of Equimatchable Graphs of Fixed Genus

A graph G is said to be equimatchable if every matching in G extends to (i.e., is a subset of) a maximum matching in G. In an earlier paper with Saito, the authors showed that there are only a finite number of 3-connected equimatchable planar graphs. In the present paper, this result is extended by showing that in a surface of any fixed genus (orientable or non-orientable), there are only a finite number of 3-connected equimatchable graphs having a minimal embedding of representativity at least three. (In fact, if the graphs considered are non-bipartite, the representativity three hypothesis may be dropped.) The proof makes use of the Gallai-Edmonds decomposition theorem for matchings.      

Combinatorics and Algorithms for Augmenting Graphs

The notion of augmenting graphs generalizes Berge’s idea of augmenting chains, which was used by Edmonds in his celebrated solution of the maximum matching problem. This problem is a special case of the more general maximum independent set (MIS) problem. Recently, the augmenting graph approach has been successfully applied to solve MIS in various other special cases. However, our knowledge of augmenting graphs is still very limited and we do not even know what the minimal infinite classes of augmenting graphs are. In the present paper, we find an answer to this question and apply it to extend the area of polynomial-time solvability of the maximum independent set problem.     

On matching and total domination in graphs

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The matching number is the maximum cardinality of a matching of G, while the total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we investigate the relationships between the matching and total domination number of a graph. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number, and we show that every k-regular graph with k?3 has total domination number at most its matching number. In general, we show that no minimum degree is sufficient to guarantee that the matching number and total domination number are comparable.     

Kernels and perfectness in arc-local tournament digraphs

In this paper we give a characterization of kernel-perfect (and of critical kernel-imperfect) arc-local tournament digraphs. As a consequence, we prove that arc-local tournament digraphs satisfy a strenghtened form of the following interesting conjecture which constitutes a bridge between kernels and perfectness in digraphs, stated by C. Berge and P. Duchet in 1982: a graph G is perfect if and only if any normal orientation of G is kernel-perfect. We prove a variation of this conjecture for arc-local tournament orientable graphs. Also it is proved that normal arc-local tournament orientable graphs satisfy a stronger form of Berge's strong perfect graph conjecture.     

A note on minimal matching covered graphs

A graph is called matching covered if for its every edge there is a maximum matching containing it. It is shown that minimal matching covered graphs without isolated vertices contain a perfect matching.     

Perfect Matchings in Balanced Hypergraphs – A Combinatorial Approach

Our topic is an extension of the following classical result of Hall to hypergraphs: A bipartite graph G contains a perfect matching if and only if for each independent set X of vertices, at least |X| vertices of G are adjacent to some vertex of X. Berge generalized the concept of bipartite graphs to hypergraphs by defining a hypergraph G to be balanced if each odd cycle in G has an edge containing at least three vertices of the cycle. Based on this concept, Conforti, Cornuéjols, Kapoor, and Vušković extended Hall's result by proving that a balanced hypergraph G contains a perfect matching if and only if for any disjoint sets A and B of vertices with |A| > |B|, there is an edge in G containing more vertices in A than in B (for graphs, the latter condition is equivalent to the latter one in Hall's result). Their proof is non-combinatorial and highly based on the theory of linear programming. In the present paper, we give an elementary combinatorial proof. Received April 29, 1997     

Perfect Graphs with No Balanced Skew‐Partition are 2‐Clique‐Colorable

A graph G is perfect if for all induced subgraphs H of G, . A graph G is Berge if neither G nor its complement contains an induced odd cycle of length at least five. The Strong Perfect Graph Theorem [9] states that a graph is perfect if and only if it is Berge. The Strong Perfect Graph Theorem was obtained as a consequence of a decomposition theorem for Berge graphs [M. Chudnovsky, Berge trigraphs and their applications, PhD thesis, Princeton University, 2003; M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann Math 164 (2006), 51–229.], and one of the decompositions in this decomposition theorem was the “balanced skew‐partition.” A clique‐coloring of a graph G is an assignment of colors to the vertices of G in such a way that no inclusion‐wise maximal clique of G of size at least two is monochromatic, and the clique‐chromatic number of G, denoted by , is the smallest number of colors needed to clique‐color G. There exist graphs of arbitrarily large clique‐chromatic number, but it is not known whether the clique‐chromatic number of perfect graphs is bounded. In this article, we prove that every perfect graph that does not admit a balanced skew‐partition is 2‐clique colorable. The main tool used in the proof is a decomposition theorem for “tame Berge trigraphs” due to Chudnovsky et al. ( http://arxiv.org/abs/1308.6444 ).     

Matching cutsets in graphs

Let G = (V,E) be an undirected graph. A subset F of E is a matching cutset of G if no two edges of F are incident with the same point, and G-F has more components than G. Chv?atal [2] proved that it is NP-complete to recognize graphs with a matching cutset even if the input is restricted to graphs with maximum degree 4. We prove the following: (a) Every connected graph with maximum degree ?3 and on more than 7 points has a matching cutset. (In particular, there are precisely two connected cubic graphs without a matching cutset). (b) Line graphs with a matching cutset can be recognized in O(|E|) time. (c) Graphs without a chordless circuit of length 5 or more that have a matching cutset can be recognized in O(|V||E|³) time.     

Matching graphs

The matching graph M(G) of a graph G is that graph whose vertices are the maximum matchings in G and where two vertices M₁ and M₂ of M(G) are adjacent if and only if |M₁ − M₂| = 1. When M(G) is connected, this graph models a metric space whose metric is defined on the set of maximum matchings in G. Which graphs are matching graphs of some graph is not known in general. We determine several forbidden induced subgraphs of matching graphs and add even cycles to the list of known matching graphs. In another direction, we study the behavior of sequences of iterated matching graphs. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 73–86, 1998     

Tutte sets in graphs II: The complexity of finding maximum Tutte sets

A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency. A subset X of V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. We explored the structural aspects of Tutte sets in another paper. Here, we consider the algorithmic complexity of finding Tutte sets in a graph. We first give two polynomial algorithms for finding a maximal Tutte set. We then consider the complexity of finding a maximum Tutte set, and show it is NP-hard for general graphs, as well as for several interesting restricted classes such as planar graphs. By contrast, we show we can find maximum Tutte sets in polynomial time for graphs of level 0 or 1, elementary graphs, and 1-tough graphs.     

On Unicyclic Graphs with Uniquely Restricted Maximum Matchings

A graph is called unicyclic if it owns only one cycle. A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. Clearly, μ _r (G) ≤ μ(G), where μ _r (G) denotes the size of a maximum uniquely restricted matching, while μ(G) equals the matching number of G. In this paper we study unicyclic bipartite graphs enjoying μ _r (G) = μ(G). In particular, we characterize unicyclic bipartite graphs having only uniquely restricted maximum matchings. Finally, we present some polynomial time algorithms recognizing unicyclic bipartite graphs with (only) uniquely restricted maximum matchings.