( P5,diamond)-free graphs revisited: structure and linear time optimization

Abstract

Using the concept of prime graphs and modular decomposition of graphs, we give a complete structure description of ( P 5,diamond)-free graphs implying that these graphs have bounded clique width (the P 5 is the induced path with five vertices a, b, c, d, e and four edges ab, bc, cd, de, and the diamond consists of four vertices a, b, c, d such that a, b, c form an induced path with edges ab, bc, and vertex d is adjacent to a, b and c). The structure and bounded clique width of this graph class allows to solve several algorithmic problems on this class in linear time, among them the problems Maximum Weight Stable Set (MWS), Maximum Weight Clique, Domination, Steiner Tree and in general every algorithmic problem which is, roughly speaking, expressible in a certain kind of Monadic Second-Order Logic using quantification only over vertex but not over edge set predicates. This improves previous results on ( P 5,diamond)-free graphs in several ways: We give a complete structure description of prime ( P 5,diamond)-free graphs, we do not only solve the MWS problem on this class, we achieve linear time algorithms (instead of a recent time bound O ( n m ) ), and we can do all this on a larger graph class containing ( P 5,diamond)-free graphs which admits linear time recognition.