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 $(\sqrt{2})^n$ minimum dominating sets (MDS), and the structure of trees containing precisely $(\sqrt{2})^n$ MDS is described. On the other hand, for $d=5$, an $n$-vertex tree containing more than $(1/3) \cdot 1.415^n$ MDS is constructed for each $n \geq 1$. It is shown that each $n$-vertex tree contains fewer than $1.4205^n$ MDS.