A new Greedoid: The family of local maximum stable sets of a forest

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set if it is a maximum stable of the subgraph of G spanned by S∪N(S), where N(S) is the neighborhood of S. One theorem of Nemhauser and Trotter Jr. (Math. Programming 8 (1975) 232-248), working as a useful sufficient local optimality condition for the weighted maximum stable set problem, ensures that any local maximum stable set of G can be enlarged to a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for forests. Namely, we show that for any non-empty local maximum stable set S of a forest T there exists a local maximum stable set S ₁ of T, such that S ₁⊂S and |S ₁|=|S|-1. Moreover, as a further strengthening of both the theorem of Nemhauser and Trotter Jr. and its inverse, we prove that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set.