Upper Domination Number and Domination Number in a Tree

Abstract

For a graph \(G = (V(G),E(G))\), a set \(S \subseteq V(G)\) is called a dominating set if \(N_G[S] = V(G)\). A dominating set \(S\) is said to be minimal if no proper subset \(S’ \subset S\) is a dominating set. Let \(\gamma(G)\) (called the domination number) and \(\Gamma(G)\) (called the upper domination number) be the minimum cardinality and the maximum cardinality of a minimal dominating set of \(G\), respectively. For a tree \(T\) of order \(n \geq 2\), it is obvious that \(1 = \gamma(K_{1,n-1}) \leq \gamma(T) \leq \Gamma(T) \leq \Gamma(K_{1,n-1}) = n-1\). Let \(t(n) = \min_{|T|=n}(\Gamma(T)-\gamma(T))\). In this paper, we determine \(t(n)\) for all natural numbers \(n\). We also characterize trees \(T\) with \(\Gamma(T) – \gamma(T) = t(n)\).