The intersection of all maximum stable sets of a tree and its pendant vertices

A stable set in a graph G is a set of mutually non-adjacent vertices, α(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T , of order n ≥ 2, any stable set of size ≥ n/2 contains at least one pendant vertex. Hence, we deduce that any maximum stable set in a tree contains at least one pendant vertex. Our main finding is the theorem claiming that if T does not own a perfect matching, then at least two pendant vertices an even distance apart belong to core(T ). While it is known that if G is a connected bipartite graph of order n ≥ 2, then |core(G)| 6= 1 (see Levit, Mandrescu [8]), our new statement reveals an additional structure of the intersection of all maximum stable sets of a tree. The above assertions give refining of one result of Hammer, Hansen and Simeone [3], stating that if a graph G is of order less than 2α(G), then core(G) is non-empty, and also of a result of Jamison [6], Gunter, Hartnel and Rall [2], and Zito [11], saying that for a tree T of order at least two, |core(T )| 6= 1.