Connected Outer-Hop Independent Dominating Sets in Graphs Under Some Binary Operations

Abstract

Let $G$ be a connected graph. A set $D\subseteq V(G)$ is called a connected outer-hop independent dominating if $D$ is a connected dominating set and $V(G)\s D$ is a hop independent set in $G$, respectively. The minimum cardinality of a connected outer-hop independent dominating set in $G$, denoted by $\gamma_{c}^{ohi}(G)$, is called the connected outer-hop independent domination number of $G$. In this paper, we introduce and investigated the concept of connected outer-hop independent domination in a graph. We show that the connected outer-hop independent domination number and connected outer-independent domination number of a graph are incomparable. In fact, we find that their absolute difference can be made arbitrarily large. In addition, we characterize connected outer-hop independent dominating sets in graphs under some binary operations. Furthermore, these results are used to give exact values or bounds of the parameter for these graphs.