Maximum Stable Sets and Pendant Vertices in Trees

One theorem of Nemhauser and Trotter [10] ensures that, under certain conditions, a stable set of a graph G can be enlarged to a maximum stable set of this graph. For example, any stable set consisting of only simplicial vertices is contained in a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for trees of order greater than one, where, in fact, all the simplicial vertices are pendant. Namely, we show that any maximum stable set of such a tree contains at least one pendant vertex. Moreover, we prove that if T does not own a perfect matching, then a stable set, consisting of at least two pendant vertices, is included in the intersection of all its maximum stable sets. For trees, the above assertion is also a strengthening of one result of Hammer et al., [3], stating that if G is of order less that 2α(G) (where α(G) is the size of a maximum stable set of G), then the intersection of all its maximum stable sets is non-empty.