A note on α-redundant vertices in graphs

A vertex v in a graph G is called α-redundant if α(G−v)=α(G), where α(G) stands for the stability number of G, i.e. the maximum size of a subset of pairwise nonadjacent vertices. We describe sufficient conditions for a vertex to be α-redundant in terms of some P4 extensions. This leads to polynomial-time algorithms for solving the stable set problem giving for an arbitrary input graph either the solution of the problem or a forbidden configuration such as an induced P5 or an induced banner in the input graph. The algorithms extend and improve a number of previous results on the problem.