2017/08/01 by Hashemipour, M., Hooshmandasl, M. R., Shakiba, A.
#Discrete Mathematics (cs.DM) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.1708.00188
An outer-connected dominating set for an arbitrary graph G is a set D ⊆ V such that D is a dominating set and the induced subgraph G [V ∖ D] be connected. In this paper, we focus on the outer-connected 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 γc(G ∘ K)γc(H ∘ K) ≤ γc(G\Box H)∘ K. Furthermore, we study the outer-connected domination number of the direct product of finitely many complete graphs.