Secure domination in degree splitting graphs of certain graphs

Abstract

A  secure dominating set of a graph $G=\left(V,E\right)$  is a dominating set $S\subseteq V$ if for each $u\in V-S$, there exists a $v$ such that $v\in N\left(u\right)\bigcap S$ and $\left(S-\left\{v\right\}\right)\bigcup \left\{u\right\}$ is dominating. The minimum cardinality of a secure dominating set is the secure domination number, ${\gamma }_s\left(G\right)$. In this paper, we find  ${\gamma }_s\left(G\right)$ for degree splitting graphs of few classes of graphs like paths, complete binary trees, complete graphs and complete bipartite graphs. Further, bounds for ${\gamma }_s\left(G\right)$ of degree splitting graphs of regular graphs and few classes of caterpillars are determined.