The number of dominating sets in d-ary trees

Abstract

An (unrooted) d-ary tree is a tree in which every internal vertex has degree \(d+1\). In this paper, we show for every fixed \(d\ge 2\) that d-ary caterpillars have the minimum number of dominating sets among d-ary trees of a given order. We also determine the maximum number of dominating sets in binary trees (the special case \(d=2\)) and classify the extremal trees, which are also unique.