Vertex-Edge Dominating Sets of Some Graphs Under Binary Operations

Abstract

Given a simple undirected graph $G=(V(G),E(G))$, a vertex $u\in V(G)$ vertex-edge dominates the edge  $xy\in E(G)$ if one of the following holds: $(1)$ $u=x$ or $u=y$, $(2)$ $ux\in E(G)$ or $uy\in E(G)$. A subset $S\subseteq V(G)$ is a vertex-edge dominating set of $G$ if for each $xy\in E(G)$, there exists $u\in S$ such that $u$ vertex-edge dominates $xy$. A vertex-edge dominating set $S\subseteq V(G)$ is a total vertex-edge dominating set if for each $u\in S$, there exists $v\in S$ for which $uv\in E(G)$. The minimum cardinality of a vertex-edge (resp. total vertex-edge) dominating set of $G$ is the vertex-edge domination number (resp. total vertex-edge domination number) of $G$. This paper investigates the vertex-edge domination and total vertex-edge domination in the join, corona, lexicographic product, complementary prism and edge corona of graphs. It provides complete characterizations of both the vertex-edge dominating sets and total vertex-edge dominating sets in these families of graphs, and establishes sharp bounds, if not the exact values, for their respective vertex-edge domination and total vertex-edge domination numbers.