On the outer-connected domination number for graph products

Abstract

An outer-connected dominating set for an arbitrary graph G is a setD ⊆ V such thatD is a dominating set and the induced subgraph G[V \D] be connected. In this paper, we focus on the outerconnected domination number of the product of graphs. We investigate the existence of outer-connected dominating set in lexicographic product and Corona of two arbitrary graphs, and we present upper bounds for outer-connected domination number in lexicographic and Cartesian product of graphs. Also, we establish an equivalent form of the Vizing's conjecture for outer-connected domination number in lexicographic and Cartesian product asγc(G • K)γc(H • K) ≤γc(G H) • K. Furthermore, we study the outer-connected domination number of the direct product of finitely many complete graphs.