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The strong perfect graph theorem
 ANNALS OF MATHEMATICS
, 2006
"... A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The “strong perfect graph conjecture” (Berge, 1961) asse ..."
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Cited by 161 (13 self)
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A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The “strong perfect graph conjecture” (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornuéjols and Vuˇsković — that every Berge graph either falls into one of a few basic classes, or admits one of a few kinds of separation (designed so that a minimum counterexample to Berge’s conjecture cannot have either of these properties). In this paper we prove both these conjectures.
The Homogeneous Set Sandwich Problem
, 1998
"... The graph sandwich problem for property \Phi is defined as follows: Given two graphs G 1 = (V; E 1 ) and G 2 = (V; E 2 ) such that E 1 ` E 2 , is there a graph G = (V; E) such that E 1 ` E ` E 2 which satisfies property \Phi? We present a polynomialtime algorithm for solving the graph sandwich pro ..."
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Cited by 7 (4 self)
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The graph sandwich problem for property \Phi is defined as follows: Given two graphs G 1 = (V; E 1 ) and G 2 = (V; E 2 ) such that E 1 ` E 2 , is there a graph G = (V; E) such that E 1 ` E ` E 2 which satisfies property \Phi? We present a polynomialtime algorithm for solving the graph sandwich problem, when property \Phi is "to contain a homogeneous set". The algorithm presented also provides the graph G and a homogeneous set in G in case it exists.
SPLITPERFECT GRAPHS: CHARACTERIZATIONS AND ALGORITHMIC USE
, 2004
"... Two graphs G and H with the same vertex set V are P4isomorphic if every four vertices {a, b, c, d} ⊆V induce a chordless path (denoted by P4) inG if and only if they induce a P4 in H. We call a graph splitperfect if it is P4isomorphicto a split graph (i.e., a graph being partitionable into a cl ..."
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Cited by 6 (2 self)
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Two graphs G and H with the same vertex set V are P4isomorphic if every four vertices {a, b, c, d} ⊆V induce a chordless path (denoted by P4) inG if and only if they induce a P4 in H. We call a graph splitperfect if it is P4isomorphicto a split graph (i.e., a graph being partitionable into a clique and a stable set). This paper characterizes the new class of splitperfect graphs using the concepts of homogeneous sets and pconnected graphs and leads to a linear time recognition algorithm for splitperfect graphs, as well as efficient algorithms for classical optimization problems on splitperfect graphs based on the primeval decomposition of graphs. The optimization results considerably extend previous ones on smaller classes such as P4sparse graphs, P4lite graphs, P4laden graphs, and (7,3)graphs. Moreover, splitperfect graphs form a new subclass of brittle graphs containing the superbrittle graphs for which a new characterization is obtained leading to linear time recognition.
A new characterization of P 4 connected graphs
, 1997
"... A graph is said to be P 4 connected if for every partition of its vertexset into two nonempty disjoint sets, some P 4 in the graph has vertices from both sets in the partition. A P 4 chain is a sequence of vertices such that every four consecutive ones induce a P 4 . The main result of this wo ..."
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Cited by 2 (1 self)
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A graph is said to be P 4 connected if for every partition of its vertexset into two nonempty disjoint sets, some P 4 in the graph has vertices from both sets in the partition. A P 4 chain is a sequence of vertices such that every four consecutive ones induce a P 4 . The main result of this work states that a graph is P 4 connected if and only if each pair of vertices is connected by a P 4 chain. Our proof relies, in part, on a lineartime algorithm that, given two disitnct vertices, exhibits a P 4 chain connecting them. In addition to shedding new light on the structure of P 4 connected graphs, our result extends a previously known theorem about the P 4 structure of unbreakable graphs. 1 Introduction Very recently, B. Jamison and S. Olariu [10] introduced the notion of P 4 connectedness. Specifically, a graph G = (V; E) is P 4 connected if for every partition of the vertexset V into nonempty disjoint sets V 1 and V 2 , some chordless path on four vertices and three...
On the P4Structure of Perfect Graphs: V. Overlap Graphs
 J. Combin. Theory Ser. B
, 1993
"... Given a graph G we define its koverlap graph as the graph whose vertices are the induced P4 's of G and two vertices in the overlap graph are adjacent if the corresponding P4 's in G have exactly k vertices in common. For k = 1; 2; 3 we prove that if the koverlap graph of G is bipartite then G ..."
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Cited by 2 (2 self)
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Given a graph G we define its koverlap graph as the graph whose vertices are the induced P4 's of G and two vertices in the overlap graph are adjacent if the corresponding P4 's in G have exactly k vertices in common. For k = 1; 2; 3 we prove that if the koverlap graph of G is bipartite then G is perfect.
Building counterexamples
 DIMACS Workshop on Perfect Graphs, Princeton
, 1993
"... A conjecture concerning perfect graphs asserts that if for a Berge graph G the following three conditions hold: 1. neither G, nor ¯ G has an even pair; 2. neither G, nor ¯ G has a stable cutset; 3. neither G, nor ¯ G has a starcutset, then G or ¯ G is diamondfree. We show that this conjecture is n ..."
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Cited by 1 (0 self)
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A conjecture concerning perfect graphs asserts that if for a Berge graph G the following three conditions hold: 1. neither G, nor ¯ G has an even pair; 2. neither G, nor ¯ G has a stable cutset; 3. neither G, nor ¯ G has a starcutset, then G or ¯ G is diamondfree. We show that this conjecture is not valid and that, in a way, every weaker version is false too. To this end, we construct a class of perfect graphs satisfying the hypothesis above and indicate counterexamples within this class for the instances of the conjecture obtained by replacing the diamond with any graph H which is the join of a clique and a stable set. 1. Introduction. For a graph G = (V,E), let us call a clique any set of pairwise adjacent vertices in G. The clique number ω(G) of G represents the cardinality of a largest clique in G, while the chromatic number χ(G) of G is the minimum number of colours necessary to colour the vertices of G in such a way that any two adjacent vertices have different colours. Using
Quasiparity and perfect graphs
 Inf. Proc. Lett
, 1995
"... In order to prove the Strong Perfect Graph Conjecture, the existence of a ”simple ” property P holding for any minimal nonquasiparity Berge graph G would really reduce the difficulty of the problem. We prove here that this property cannot be of type ”G is Ffree”, where F is any fixed family of Be ..."
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Cited by 1 (0 self)
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In order to prove the Strong Perfect Graph Conjecture, the existence of a ”simple ” property P holding for any minimal nonquasiparity Berge graph G would really reduce the difficulty of the problem. We prove here that this property cannot be of type ”G is Ffree”, where F is any fixed family of Berge graphs.
Polynomial Time Recognition of P 4 structure
"... A P4 is a set of four vertices of a graph that induces a chordless path; the P4structure of a graph is the set of all P4 's. Vasek Chvatal asked if there is a polynomial time algorithm to determine whether an arbitrary fouruniform hypergraph is the P4structure of some graph. The answer is yes; we ..."
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A P4 is a set of four vertices of a graph that induces a chordless path; the P4structure of a graph is the set of all P4 's. Vasek Chvatal asked if there is a polynomial time algorithm to determine whether an arbitrary fouruniform hypergraph is the P4structure of some graph. The answer is yes; we present such an algorithm.
Perfect graphs with unique P4structure
, 1996
"... We will extend Reed's SemiStrong Perfect Graph Theorem by proving that unbreakable C 5 free graphs different from a C 6 and its complement have unique P 4 structure. ..."
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We will extend Reed's SemiStrong Perfect Graph Theorem by proving that unbreakable C 5 free graphs different from a C 6 and its complement have unique P 4 structure.