On the Number of Minimum Dominating Sets in Trees

The class of trees in which the degree of each vertex does not exceed an integer d is considered. It is shown that, for d = 4, each n-vertex tree in this class contains at most (root 2)(n) minimum dominating sets (MDS), and the structure of trees containing precisely (v2)(n) MDS is described. On the other hand, for d = 5, an n-vertex tree containing more than (1/3) . 1.415(n) MDS is constructed for each n >= 1. It is shown that each n-vertex tree contains fewer than 1.4205(n) MDS.