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

A stable set in a graph image is a set of mutually non-adjacent vertices, image is the size of a maximum stable set of image, and image is the intersection of all its maximum stable sets. It is known that if image is a connected graph of order image with image, then image, [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 image is a tree of order image, with image, then at least two pendant vertices an even distance apart belong to image.