(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Given a simple undirected graph G=(V(G),E(G)), a vertex uV(G) vertex-edge dominates the edge xyE(G) if one of the following holds: (1) u=x or u=y, (2) uxE(G) or uyE(G). A subset SV(G) is a vertex-edge dominating set of G if for each xyE(G), there exists uS such that u vertex-edge dominates xy. A vertex-edge dominating set SV(G) is a total vertex-edge dominating set if for each uS, there exists vS for which uvE(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.