Secure Pointwise Non-Domination and Secure Hop Domination in Graphs

Abstract

In this paper, we revisit the concept of secure hop domination in graphs and define a new concept called secure pointwise non-domination. A pointwise non-dominating set $S$ is a secure pointwise non-dominating set if for every $u \in V (G) \setminus S$, there exists $v \in S \setminus N_G(u)$ such that $(S \setminus {v}) \cup {u}$ is a pointwise non-dominating set. The secure pointwise non-domination number $spnd(G)$ of $G$ is the smallest cardinality of a secure pointwise non-dominating set in $G$. In this paper, we give bounds on the secure pointwise non-domination number and characterize those graphs which attain these bounds. We also determine the secure pointwise non-domination number of some classes of graphs. Necessary and sufficient conditions for a subset in the join of graphs to be a secure hop dominating set is given. Moreover, we show that given positive integers $a$ and $b$ with $2 \leq a \leq b$, there exists a connected graph such that $\gamma_h(G) = a$ and $\gamma_sh(G) = b$, where $\gamma_h(G)$ and $\gamma_sh(G)$ are the hop domination number and secure hop domination number of $G$, respectively.