Inverse Complementary Tree Domination Number of Total Graphs of $P_n$ and $C_n$

Abstract

A non-empty set $D \subseteq V$ of a graph is a dominating set if every vertex in $V-D$ is adjacent to some vertex in $D$. The domination number $\gamma(G)$ of $G$ is the minimum cardinality taken over all the minimal dominating sets of $G$. A dominating set $D$ is called a complementary tree dominating set if the induced subgraph $\langle V-D\rangle$ is a tree. The complementary tree domination number $\gamma_{ctd}(G)$ of $G$ is the minimum cardinality taken over all minimal complementary tree dominating sets of $G$. Let $D$ be a minimum dominating set of $G$. If $V-D$ contains a dominating set $D^\prime$, then $D^\prime$ is called the inverse dominating set of $G$ w.r.t to $D$. The inverse domination number $\gamma^\prime(G)$ is the minimum cardinality taken over all the minimal inverse dominating sets of $G$. In this paper, inverse complementary tree domination in total graphs of $P_n$ and $C_n$ are obtained.