Characterization of Total Very Excellent Trees

Abstract

Let $G = (V,E)$ be a simple graph. A subset D of V is said to be a total dominating set of G if every vertex $v \in V$ is adjacent to at least one vertex of D. The total domination number $\gamma _t(G)$ is the minimum cardinality of a total dominating set of G. A total dominating set with $\gamma _t(G)$ cardinality is said to be a $\gamma _t$-set of G. A graph G is said to be total excellent if given any vertex x of G, there is a $\gamma _t(G)$-set of G containing x. A $\gamma _t$-set D of G is said to be total very excellent $\gamma _t$-set of G if for each vertex $u \in V - D$, there is a vertex $v \in D$ such that $(D - v) \cup \{u\}$ is a $\gamma _t$-set of G. The graph G is said to be total very excellent if it has at least one total very excellent $\gamma _t$-set. Total very excellent graphs are total excellent. In this paper we characterize total very excellent caterpillars and total very excellent trees.