Double Edge-Vertex Domination on Middle and Splitting Graphs of Path and Cycle

Abstract

An edge $e=uv$ of graph $G=(V,E)$ is said to be edge-vertex dominate vertices $u$ and $v$, as well as all vertices adjacent to $u$ and $v$. A set $S \subseteq E $ is a double edge-vertex dominating set if every vertex of $V$ is edge-vertex dominated by at least two edges of $S$. The minimum cardinality of a double edge-vertex dominating set of $G$ is the double edge-vertex domination number and is denoted by $\gamma_{dev}(G)$. In this paper, we present results for middle graphs of path and cycle and some splitting graphs of path and cycle on double edge-vertex domination numbers.