On algorithms for ( [formula omitted],gem)-free graphs

Abstract

A graph is ( P 5 ,gem)-free, when it does not contain P 5 (an induced path with five vertices) or a gem (a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path P 4 ) as an induced subgraph. We present O ( n 2 ) time recognition algorithms for chordal gem-free graphs and for ( P 5 ,gem)-free graphs. Using a characterization of ( P 5 ,gem)-free graphs by their prime graphs with respect to modular decomposition and their modular decomposition trees [A. Brandstädt, D. Kratsch, On the structure of ( P 5 ,gem)-free graphs, Discrete Appl. Math. 145 (2005), 155–166], we give linear time algorithms for the following NP-complete problems on ( P 5 ,gem)-free graphs: Minimum Coloring; Maximum Weight Stable Set; Maximum Weight Clique; and Minimum Clique Cover.