On outer-connected domination for graph products

An outer-connected dominating set for an arbitrary graph G is a set D̃ ⊆ V such that D̃ is a dominating set and the induced subgraph G[V \ D̃] 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 as γ̃c(G ◦K)γ̃c(H ◦K) ≤ γ̃c(G H) ◦K. Furthermore, we study the outer-connected domination number of the direct product of finitely many complete graphs.