Perfect Graphs, Partitionable Graphs and Cutsets

A graph G is perfect if, for all induced subgraphs of G, the size of a largest clique is equal to the chromatic number. A graph is minimally imperfect if it is not perfect but all its proper induced subgraphs are. A hole is a chordless cycle of length at least four. The strong perfect graph conjecture of Berge [1] states that G is minimally imperfect if and only if G or its complement is an odd hole. (The complement Ḡ of G is a graph with same node set as G and two nodes are adjacent in Ḡ if and only if they are not adjacent in G). A graph G is partitionable if there exist integers α and ω greater than one such that G has exactly αω+1 nodes and, for each node v∈V , G\v can be partitioned into both α cliques of size ω and ω stable sets of size α. Lovász [13] has proven that every minimally imperfect graph is partitionable.