Structure and stability number of chair-, co-P- and gem-free graphs revisited

Abstract

The P 4 is the induced path with vertices a, b, c, d and edges ab, bc, cd. The chair (co-P, gem) has a fifth vertex adjacent to b ( a and b, a, b, c and d, respectively). We give a complete structure description of prime chair-, co-P- and gem-free graphs which implies bounded clique width for this graph class. It is known that this has some nice consequences; very roughly speaking, every problem expressible in a certain kind of Monadic Second Order Logic (quantifying only over vertex set predicates) can be solved efficiently for graphs of bounded clique width. In particular, we obtain linear time for the problems Vertex Cover, Maximum Weight Stable Set (MWS), Maximum Weight Clique, Steiner Tree, Domination and Maximum Induced Matching on chair-, co-P- and gem-free graphs and a slightly larger class of graphs. This drastically improves a recently published O( n 4) time bound for Maximum Stable Set on butterfly-, chair-, co-P- and gem-free graphs.