Broadcast domination of lexicographic and modular products of graphs

Abstract

A dominating broadcast labeling of a graph G is a function (Formula presented.) such that (Formula presented.) for all (Formula presented.) where e(v) is the eccentricity of v, and for every vertex (Formula presented.) there exists a vertex v with (Formula presented.) and (Formula presented.) The cost of f is (Formula presented.) The minimum of costs over all the dominating broadcast labelings of G is called the broadcast domination number (Formula presented.) of G. In this paper, we give bounds to the broadcast domination number of lexicographic product G • H of a connected graph G and a graph H, and we show that the bounds are tight by determining the exact values for lexicographic products of some classes of graphs. Also, we give an algorithm which produces a dominating broadcast labeling of G • H. We use the algorithm to find (Formula presented.) • (Formula presented.) and (Formula presented.) • (Formula presented.) Further, we give an upper bound for the broadcast domination number of the modular product of two connected graphs and exact values are determined for the modular product of two graphs involving path, cycle and complete graphs. © 2022 The Author(s). Published with license by Taylor & Francis Group, LLC.