New sufficient conditions for αα-redundant vertices

A vertex vv in a graph GG is called αα-redundant   if α(G−v)=α(G)α(G−v)=α(G), where α(G)α(G) stands for the independence number of GG, i.e. the maximum size of a subset of pairwise non-adjacent vertices. We will recall some results about αα-redundant vertices and show some new sufficient conditions for a vertex to be αα-redundant. Based on this, we will give a unified view about vertex removal techniques for solving the maximum independent set problem. It also leads to an efficient way to solve the problem in some subclasses of S1,2,2S1,2,2-free graphs and S2,2,2S2,2,2-free graphs, where Si,j,kSi,j,k is the graph consisting of three induced paths of lengths ii, jj and kk with a common initial vertex.