A new lower bound on the domination number of a graph

Abstract

A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number, $\gamma (G)$, of G is the minimum cardinality of a dominating set of G. Lemańska (Discuss Math Graph Theory 24:165–170, 2004) showed that if T is a tree of order $n \ge 2$ with $\ell$ leaves, then $\gamma (T) \ge (n-\ell +2)/3$, and characterized all trees achieving equality in this bound. In this paper, we first characterize all trees T of order n with $\ell$ leaves satisfying $\gamma (T) = \lceil (n - \ell + 2)/3 \rceil$. We then generalize this result to connected graphs and show that if G is a connected graph of order $n \ge 2$ with $k \ge 0$ cycles and $\ell$ leaves, then $\gamma (G) \ge \lceil (n-\ell +2 - 2k)/3 \rceil$. We also characterize the graphs G achieving equality for this new bound.