Strong Perfect Graph Research Articles

Bull-Reducible Berge Graphs are Perfect

Berge's well known SPGC (Strong Perfect Graph Conjecture) states that the class of perfect graphs coincides with the class of graphs containing no induced odd cycle of length at least 5 or the complement of such a cycle. A graph in this second class is called Berge. A bull is a graph with five vertices x, a, b, c, d and five edges xa, xb, ab, ad, bc. A graph is bull-reducible if no vertex is in two bulls. We prove that every bull-reducible Berge graph is perfect and we exhibit a polynomial-time recognition algorithm for bull-reducible Berge graphs.

On kernel-less clique-acyclic orientations of minimally imperfect graphs

In this paper we will show some properties of the kernel-less clique-acyclic orientations of minimally imperfect graphs and discuss some relation between these properties, Berge and Duchet's Conjecture about kernel-solvability and the Strong Perfect Graph Conjecture. Especially, we will give a new equivalent formalization of the SPGC by using the concept unique fractional kernel.

On Tucker's proof of the strong perfect graph conjecture for [formula omitted]-free graphs

In this note, the authors generalize the ideas presented by Tucker in his proof of the Strong Perfect Graph Conjecture for (K 4−e) -free graphs in order to find a vertex v in G whose special neighborhood allows to extend a ω(G)-vertex coloring of G−v to a ω(G)-vertex coloring of G. The search for such a vertex led us to the definition of p-Tucker vertices: vertices contained in at most two maximal cliques of size at least p; and to a family of classes G p of graphs G whose maximal cliques of size at least p have no edge in common with any other maximal cliques of G. We prove that every hole-free graph G in G p has a p-Tucker vertex and we use this fact to compute ω(G) in polynomial time for each class G p . We state a conjecture whose validity yields the validity of the Strong Perfect Graph Conjecture for (K 5−e) -free graphs clique.

Almost Perfect Matrices and Graphs

We introduce the notions of ω-projection and κ-projection that map almost integral polytopes associated with almost perfect graphs G with n nodes from ℝn into ℝn−ω, where ω is the maximum clique size in G. We show that C. Berge's strong perfect graph conjecture is correct if and only if the projection (of either kind) of such polytopes is again almost integral in ℝn−ω. Several important properties of ω-projections and κ-projections are established. We prove, among other results, that the strong perfect graph conjecture is wrong if an ω-projection and a related κ-projection of an almost integral polytope with 2 ≤ ω ≤ (n − 1)/2 produce different polytopes in ℝn−ω.

P_4-Colorings and P_4-Bipartite Graphs

A vertex partition of a graph into disjoint subsets V_is is said to be a P_4-free coloring if each color class V_i induces a subgraph without chordless path on four vertices (denoted by P_4). Examples of P_4-free 2-colorable graphs (also called P_4-bipartite graphs) include parity graphs and graphs with ''few'' P_4s like P_4-reducible and P_4-sparse graphs. We prove that, given k≥ 2, \emphP_4-Free k-Colorability is NP-complete even for comparability graphs, and for P_5-free graphs. We then discuss the recognition, perfection and the Strong Perfect Graph Conjecture (SPGC) for P_4-bipartite graphs with special P_4-structure. In particular, we show that the SPGC is true for P_4-bipartite graphs with one P_3-free color class meeting every P_4 at a midpoint.

Cayley partitionable graphs

Abstract In this paper we investigate the class of Cayley partitionable graphs. This investigation is motivated by the Strong Perfect Graph Conjecture. Cayley partitionable graphs are Cayley Graphs which are closely related to near-factorizations of finite groups. We prove some structural properties of near-factorizations and give examples of Cayley partitionable graphs which are not generated by all constructions of partitionable graphs known so far.

Forced color classes, intersection graphs and the strong perfect graph conjecture

In 1996, A. Sebő[11] raised the following two conjectures concerned with the famous Strong Perfect Graph Conjecture: (1) Suppose that a minimally imperfect graph G has a vertex p incident to 2ω(G)−2 determined edges and that its complement G ̄ has a vertex q incident to 2α(G)−2 determined edges. ( An edge of G is called determined if an ω- clique of G contains both of its endpoints.) Then G is an odd hole or an odd antihole. (2) Let v 0 be a vertex of a partitionable graph G. And suppose A,B to be ω- cliques of G so that v 0 ∈ A ∩ B . If every ω- clique K containing the vertex v 0 is contained in A ∪ B , then G is an odd hole or an odd antihole. In this paper, we will prove (1) for a minimally imperfect graph G such that (p,q) is a determined edge of either G or G ̄ , and prove (2) for a minimally imperfect graph G such that G ̄ is C 4 -free and edges of G ̄ are all determined edges.

Recognizing Perfect 2-Split Graphs

A graph is a split graph if its vertices can be partitioned into a clique and a stable set. A graph is a k-split graph if its vertices can be partitioned into k sets, each of which induces a split graph. We show that the strong perfect graph conjecture is true for 2-split graphs and we design a polynomial algorithm to recognize a perfect 2-split graph.

Some classes of strongly perfect graphs

A graph is perfect if for each of its induced subgraphs H, the chromatic number of H is equal to the maximum number of mutually adjacent vertices in it. A graph is ‘strongly perfect’ if each of its induced subgraphs H contains an independent set which meets all the cliques (maximal complete subgraphs) in it. Every strongly perfect graph is perfect but the converse is generally not valid. For example, the complement of an even cycle of length at least 6 is not strongly perfect though it is perfect. The strongly perfect graphs form an interesting class of perfect graphs, because the complement of a strongly perfect graph is not necessarily strongly perfect, unlike the case with the perfect graphs and their relevance to the famous Berge's strong perfect graph conjecture which has been eluding a solution for more than three decades. Meyniel graphs, line graphs that are free from some graphs, comparability graphs, costrongly perfect graphs are some of the most important classes of strongly perfect graphs. Here, we summarize the results concerning strongly perfect graphs.

Recognizing the P4-structure of bipartite graphs

The P 4-structure of a graph G=(V, E) is a hypergraph H=(V, E) such that the hyperedges from H correspond to the vertex sets of the induced P 4s in G. The Semi Strong Perfect Graph Theorem states that a graph is perfect if and only if it has the P 4-structure of a perfect graph. While at present no polynomial-time algorithm is known to recognize the P 4-structure of perfect graphs, first results have been obtained for special subclasses of perfect graphs. In this note we present an algorithm which decides efficiently whether a given hypergraph represents the P 4-structure of a bipartite graph.

On minimal imperfect graphs without induced P5

In this paper, we summarize previous known results on P 5-free minimal imperfect graphs (i.e. minimal imperfect graphs not containing a path on 5 vertices as induced subgraph) and we introduce two new classes of graphs (defined by a local property) that contain P 5-free graphs. Next, we show that most of the results concerning P 5-free graphs can be extended to these classes. Moreover, we present a structural characterization of these graphs which leads to some new results. In particular, we prove that the Strong Perfect Graph Conjecture holds true for P 5-free and F-free graphs where F is any connected configuration on 5 vertices not containing an induced 2 K 2.

Even and odd holes in cap-free graphs

It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge's strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is β-perfect. In this article, we extend the problem of testing whether G(V, E) contains a hole of a given parity to the case where each edge of G has a label odd or even. A subset of E is odd (resp. even) if it contains an odd (resp. even) number of odd edges. Graphs for which there exists a signing (i.e., a partition of E into odd and even edges) that makes every triangle odd and every hole even are called even-signable. Graphs that can be signed so that every triangle is odd and every triangle is odd and every hole is odd are called odd-signable. We derive from a theorem due to Truemper co-NP characterizations of even-signable and odd-signable graphs. A graph is strongly even-signable if it can be signed so that every cycle of length ≥ 4 with at most one chord is even and every triangle is odd. Clearly a strongly even-signable graph is even-signable as well. Graphs that can be signed so that cycles of length four with one chord are even and all other cycles with at most one chord are odd are called strongly odd-signable. Every strongly odd-signable graph is odd-signable. We give co-NP characterizations for both strongly even-signable and strongly odd-signable graphs. A cap is a hole together with a node, which is adjacent to exactly two adjacent nodes on the hole. We derive a decomposition theorem for graphs that contain no cap as induced subgraph (cap-free graphs). Our theorem is analogous to the decomposition theorem of Burlet and Fonlupt for Meyniel graphs, a well-studied subclass of cap-free graphs. If a graph is strongly even-signable or strongly odd-signable, then it is cap-free. In fact, strongly even-signable graphs are those cap-free graphs that are even-signable. From our decomposition theorem, we derive decomposition results for strongly odd-signable and strongly even-signable graphs. These results lead to polynomial recognition algorithms for testing whether a graph belongs to one of these classes. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 289–308, 1999

On minimal imperfect graphs with circular symmetry

Results of Lovász and Padberg entail that the class of so-called partitionable graphs contains all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture, which asserts that the only minimal imperfect graphs are the odd chordless cycles with at least five vertices (“odd holes”) and their complements (“odd antiholes”). Only two constructions (due to Chvátal, Graham, Perold, and Whitesides) are known for making partitionable graphs. The first one does not produce any counterexample to Berge's Conjecture, as shown by Sebö. Here we prove that the second one does not produce any counterexample either. This second construction is given by near-factorizations of cyclic groups based on the so-called “British number systems” introduced by De Bruijn. All the partitionable graphs generated by this second construction (in particular odd holes and odd antiholes) have circular symmetry. No other partitionable graph with circular symmetry is known, and we conjecture that none exists; in this direction we prove that any counterexample must contain a clique and a stable set with at least six vertices each. Also we prove that every minimal imperfect graph with circular symmetry must have an odd number of vertices. © 1998 John Wiley & Sons, Inc. J Graph Theory 29: 209–225, 1998

On minimal imperfect graphs with circular symmetry

Results of Lovász and Padberg entail that the class of so-called partitionable graphs contains all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture, which asserts that the only minimal imperfect graphs are the odd chordless cycles with at least five vertices (“odd holes”) and their complements (“odd antiholes”). Only two constructions (due to Chvátal, Graham, Perold, and Whitesides) are known for making partitionable graphs. The first one does not produce any counterexample to Berge's Conjecture, as shown by Sebö. Here we prove that the second one does not produce any counterexample either. This second construction is given by near-factorizations of cyclic groups based on the so-called “British number systems” introduced by De Bruijn. All the partitionable graphs generated by this second construction (in particular odd holes and odd antiholes) have circular symmetry. No other partitionable graph with circular symmetry is known, and we conjecture that none exists; in this direction we prove that any counterexample must contain a clique and a stable set with at least six vertices each. Also we prove that every minimal imperfect graph with circular symmetry must have an odd number of vertices. © 1998 John Wiley & Sons, Inc. J Graph Theory 29: 209–225, 1998

On essential components and critical sets of a graph

We introduce some new concepts, which generalize the concepts of critical edge and critical component, and investigate their relation with α-coverings of a graph, perfect graphs and the Strong Perfect Graph Conjecture. Most of the obtained results can be viewed either as analogues or generalizations of well-known properties of holes and (α, ω)-graphs. A number of open problems is also suggested.

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 counterexample to the bold conjecture

A pair of vertices (x,y) of a graph G is an o-critical pair if o(G + xy) > o(G), where G + xy denotes the graph obtained by adding the edge xy to G and o(H) is the clique number of H. The o-critical pairs are never edges in G. A maximal stable set S of G is called a forced color class of G if S meets every o-clique of G, and o-critical pairs within S form a connected graph. In 1993, G. Bacso raised the following conjecture which implies the famous Strong Perfect Graph Conjecture: If G is a uniquely o-colorable perfect graph, then G has at least one forced color class. This conjecture is called the Bold Conjecture. Here we show a simple counterexample to it. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 165–168, 1997

A characterization of normal fraternally orientable perfect graphs

While the famous Berge's Strong Perfect Graph Conjecture remains a major unsolved problem in Graph Theory, the following alternative characterization of perfect graphs was conjectured in 1982 by C. Berge and P. Duchet: A graph G is perfect if and only if any normal orientation of G is kernel-perfect. In this paper I prove the validity of a version of this conjecture for graphs which accept a normal fraternal orientation.

On transversals in minimal imperfect graphs

Chvátal (1984) proved that no minimal imperfect graph has a small transversal, that is, a set of vertices of cardinality at most α + ω − 1 which meets every ω-clique and every α-stable set. In this paper we prove that a slight generalization of this notion of small transversal leads to a conjecture which is as strong as Berge's strong perfect graph conjecture for a very large class of graphs, namely for those graphs whose diameter does not exceed 6.

On the structure of bull-free perfect graphs

A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle. Chvatal and Sbihi showed that the Strong Perfect Graph Conjecture holds for bull-free graphs. We show that bull-free perfect graphs are quasi-parity graphs, and that bull-free perfect graphs with no antihole are perfectly contractile. Our proof yields a polynomial algorithm for coloring bull-free strict quasi-parity graphs