arXiv:math/9912048v1 [math.CO] 6 Dec 1999 Maximum Stable Sets and Pendant Vertices in Trees Vadim E. Levit and Eugen Mandrescu Department of Computer Science Holon Academic Institute of Technology 52 Golomb Str., P.O. Box 305 Holon 58102, ISRAEL {levitv, eugen m}@barley.cteh.ac.il Abstract 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. 1 Introduction Throughout this paper G =(V,E) is a simple (i.e., a finite, undirected, loopless and without multiple edges) graph with vertex set V = V (G) and edge set E = E(G). If X ⊂ V , then G[X ] is the subgraph of G spanned by X . By G − W we mean the subgraph G[V − W ] , if W ⊂ V (G). We also denote by G − F the partial subgraph of G obtained by deleting the edges of F , for F ⊂ E(G), and we use G − e, if W = {e}. A stable set of maximum size will be referred as to a maximum stable set of G, and the stability number of G, denoted by α(G), is the cardinality of a maximum stable set in G. Let Ω(G) stand for the set {S : S is a maximum stable set of G}, and core(G)= ∩{S : S ∈ Ω(G)}, [8]. A graph G is called α + -stable if α(G + e)= α(G), for any edge e ∈ E( G), where G is the complement of G, [2]. The neighborhood of a vertex v ∈ V is the set N (v)= {w : w ∈ V and vw ∈ E}, while the close neighborhood of v ∈ V is N [v]= N (v) ∪{v}. For A ⊂ V , we denote N (A)= {v ∈ V − A : N (v) ∩ A = ∅}. If G[N (v)] is a complete subgraph in G, then v is a simplicial vertex of G. In particular, if |N (v)| = 1, then v is a pendant 1