On Planar Quasi-Parity Graphs

A graph G is strict quasi parity (SQP) if every induced subgraph of G that is not a clique contains a pair of vertices with no odd chordless path between them (an even pair). Hougardy conjectured that the minimal forbidden subgraphs for the class of SQP graphs are the odd chordless cycles, the complements of odd or even chordless cycles, and some line-graphs of bipartite graphs. Here we prove this conjecture for planar graphs. We also give a constructive characterization of all the planar minimal forbidden subgraphs for the class of SQP graphs. 1. Introduction. A graph G is perfect if, for every induced subgraph H of G, the chromatic number of H is equal to the size of its largest clique. A hole is any chordless cycle of length at least five. A hole is odd or even according to its length. The most famous problem in the context of perfect graphs was Berge's perfect graph conjecture: A graph G is perfect if and only if neither G nor ¯ G contains an odd hole. This was proved by Chudnovsky et al. [1]. However, other problems concerning perfect graphs remain open, some of which are related to the concept of an even pair. Two nonadjacent vertices of a graph G form an even pair (resp., odd pair) if every chordless path between them has an even (resp., odd) number of edges. A graph is strict quasi parity (SQP) if each of its induced subgraphs either contains an even pair or is a clique. Fonlupt and Uhry [3] introduced even pairs and showed that the graph obtained by contracting an even pair in a perfect graph is perfect. Meyniel also showed that no minimal imperfect graph contains an even pair [11]. He proposed the definition of SQP graphs, which, by his result, are perfect. The main interest of this class lies in the fact that many classical families of perfect graphs are SQP, as proved by Meyniel (see also the more recent survey [2]). Meyniel also suspected that the minimal non-SQP graphs must have a simple structural characterization. More formally, Hougardy [5] proposed the following conjecture. Conjecture 1. Every minimal non-SQP graph is either an odd hole, or the complement of a hole, or the line graph of a bipartite graph. Let us call obstruction any graph G which is not a clique, has no even pair, and is such …