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. It is known that if G is a connected graph of order n ≥ 2 with 2 α (G) > n, then | core (G) | ≥ 2, [V.E. Levit, E. Mandrescu, Combinatorial properties of the family of maximum stable sets of a graph, Discrete Applied Mathematics 117 (2002) 149-161; E. Boros, M.C. Golumbic, V.E. Levit, On the number of vertices belonging to all maximum stable sets of a graph, Discrete Applied Mathematics 124 (2002) 17-25]. When we restrict ourselves to the class of trees, we add some structural properties to this statement. Our main finding is the theorem claiming that if T is a tree of order n ≥ 2, with 2 α (T) > n, then at least two pendant vertices an even distance apart belong to core (T).