On the outer-connected domination number for graph products

Abstract

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 outer-connected domination number of the product of graphs. We investigate the outer-connected dominating set in the lexicographic product of two arbitrary graphs, and we present upper bounds for outer-connected domination number in the lexicographic and the Cartesian product of graphs. We establish an equivalent form of the Vizing’s conjecture for outer-connected domination number in the lexicographic and the Cartesian product as . We also show that the outer-connected domination number of the Corona product of a connected graph G of order n and some non-trivial graph H equals n times the domination number of H. Furthermore, we study the outer-connected domination number of the direct product of finitely many complete graphs.