Global dominating broadcast in graphs

Abstract

A broadcast on a graph [Formula: see text] is a function [Formula: see text] such that for each [Formula: see text], where [Formula: see text] denotes the diameter of [Formula: see text] and [Formula: see text] denotes the eccentricity of [Formula: see text]. The cost of a broadcast is the value [Formula: see text]. In this paper, we define and study a new invariant of broadcasts in graphs, which is global dominating broadcast. A dominating broadcast [Formula: see text] of a graph [Formula: see text] is a global dominating broadcast if [Formula: see text] is also a dominating broadcast of [Formula: see text] the complement of [Formula: see text]. We begin by determining the global broadcast domination number of bipartite graphs, paths, cycles, grid graphs and trees. Then we establish lower and upper bounds on the global broadcast domination number of a graph. Finally, we establish relationships between the global broadcast domination number and other parameters.