Quasi-star-cutsets and some consequences

Abstract

This paper introduces a structural property of minimal imperfect graphs, no minimal imperfect graph has a quasi-star-cutset, before describing some consequences. First, a large class of perfect graphs is obtained from the class of all bipartite graphs and their line-graphs; this class contains several well-known classes of perfect graphs. The second consequence deals with a composition operation of graphs which preserves perfection and which encompasses the most part of the compositions known to preserve perfection.

Introduction

A graph G is perfect if for every induced subgraph H of G, the chromatic number χ(H) of H equals the size ω(H) of a largest clique of H. Background results on perfect graphs can be found in Berge and Chvátal [2], or in Golumbic [11]. A graph is minimal imperfect if it is not perfect but all its proper induced subgraphs are perfect. Berge [1] conjectured that a graph is perfect if and only if it has no odd hole (odd chordless cycles of length at least five) and no odd antiholes (complements of odd holes). This is nowadays known as the strong perfect graph conjecture. Berge also conjectured, and Lovász proved [16], that a graph G is perfect if and only if its complement $\text{G}$ is. This result, known as the perfect graph theorem, is equivalent to the following: a graph is minimal imperfect if and only if its complement is. In order to elucidate the structure of minimal imperfect graphs several properties of minimal imperfect graphs have been found. Two of these properties were settled by Chvátal [5] and Meyniel [17]. A few definitions are needed for describing them. In a graph G, a set C of vertices is called a cutset if G–C is disconnected. A star-cutset is a cutset C having a vertex adjacent to all other vertices of C. Two nonadjacent vertices in a graph G are called an even pair (resp. odd pair) if any chordless path between these vertices has an even (resp. odd) number of edges.

Star-cutset lemma [5]

No minimal imperfect graph has a star-cutset.

The notion of a star-cutset was also introduced by Tucker [24].

Even pair lemma [17]

No minimal imperfect graph contains an even pair.

This last property led Meyniel and Olariu [18] to propose the following conjecture.

Odd pair conjecture

No minimal imperfect graph contains an odd pair.

An equivalent version of the odd pair conjecture which is still unresolved can be seen in [15] or in [12]. A weaker version of this conjecture was proven by Hoàng [12], where some other variations of the odd pair conjecture are also suggested: no minimal imperfect graph contains an odd pair {u,v} such that all chordless paths between the vertices u and v have exactly three edges.

The important concept of a star-cutset is underlying several perfection-preserving operations like the clique identification [2], [11], the amalgam [3] and the connecting clique system [9] which were used in recognizing Meyniel graphs [3], perfect (K₄−e)-free graphs [10] and dart-free perfect graphs [8].

The main purpose of this paper is to present a structural property for minimal imperfect graphs, the quasi-star-cutset lemma, which generalizes the star-cutset lemma and which will be underlying a wide range of perfection-preserving operations. It also sets out some consequences which, on the one hand, point up a large class of perfect graphs, and on the other hand, unify several perfection-preserving compositions of graphs that seemed distant the ones from the others. The first consequence uses also the following lemma:

Odd pair cutset lemma

No minimal imperfect graph has an odd pair as a cutset.

The odd pair cutset lemma can be proven by using either a result of Tucker [23] or a result of Sebö [20] or otherwise.

What we call a quasi-star-cutset in a graph G is a cutset C=A∪B such that the two following conditions hold:

• (i) A is nonempty and every pair of nonadjacent vertices of A is an even pair in G;

• (ii) each vertex of A is adjacent to each vertex of B.

A star-cutset is a quasi-star-cutset C=A∪B with |A|=1. We are now in position to state the following:

Quasi-star-cutset lemma

No minimal imperfect graph has a quasi-star-cutset.

The quasi-star-cutset lemma can be proven by combining the star-cutset lemma and the even pair lemma, or as this can be verified, by restating the same arguments used by Chvátal for proving the star-cutset lemma. Hence, the proof of the quasi-star-cutset lemma will be omitted, as the proof of the odd pair cutset lemma.

Moreover, Chvátal [5] defined a skew partition in a graph G as a partition of the set of the vertices of G into two nonempty disjoint parts such that the first part induces a disconnected subgraph in G and the second part induces a disconnected subgraph in the complement of G. He also noticed that, for graphs with at least five vertices and at least one edge, having a star-cutset implies having a skew partition. And he made the skew partition conjecture (no minimal imperfect graph has a skew partition) which generalizes the star-cutset-lemma but not the quasi-star-cutset lemma. As it can be seen with the graph G=(V,E), hereafter defined, which is neither bipartite nor a line-graph of a bipartite graph and which has a quasi-star-cutset C={1,5} but neither a star-cutset nor a skew partition: the nodes of V are numbered from 1 to 14 and E={(1,2), (2,3), (3,4), (4,5), (5,6), (6,7), (7,8), (8,1), (1,9), (9,10), (10,11), (11,12), (12,13), (13,14), (14,9), (14,1), (5,11), (5,12)}.

The next section is devoted to the first consequence of the quasi-star-cutset lemma. A class of perfect graphs, called $\text{BLB}$, is defined and it is shown that it contains several well-known classes of perfect graphs: $\text{Bip}{}^{\mathrm{\ast }}$, perfect (K₄−e)-free graphs and dart-free perfect graphs. Section 3 outlines the second consequence which deals with a composition operation of graphs that preserves perfection and that encompasses the most part of the compositions known to preserve perfection. There, it is also proved that some subclasses of the class of strict quasi-parity graphs [17] are stable for this composition operation.

Finally, let us observe that the notion of a quasi-star-cutset is general enough to neither let possible to get a polynomial-time algorithm for testing membership in $\text{BLB}$ nor let possible to decide in polynomial time if an arbitrary graph has a quasi-star-cutset. Both of these questions must be examined in special classes of graphs and for special quasi-star-cutsets, but not in the general case.