Complementary Tree Domination in Unicyclic Graphs

Abstract

A set $D$ of a graph $G = (V, E)$ is a dominating set of every vertex in $V-D$ is adjacent to some vertex in $D$. The domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set. A dominating set $D$ is called a complementary tree dominating set if the induced subgraph $ $ is a tree. The minimum cardinality of a complementary tree dominating set is called the complementary tree domination number of $G$ and is denoted by $\gamma_{ctd}(G)$. In this paper, connected unicyclic graphs for which $\gamma_{ctd}(G) = \gamma(G)$ nad $\gamma_{ctd}(G) = \gamma(G) + 1$ are characterized.