Dominating and Irredundant Broadcasts in Graphs

TL;DR

This paper explores the properties of dominating and irredundant broadcasts in graphs, establishing bounds between their minimum costs and analyzing their relationships and ratios.

Contribution

It introduces the concept of irredundant broadcasts, determines conditions for maximal irredundance, and establishes bounds between various broadcast parameters in graphs.

Findings

ir_b(G) is bounded above by gamma_b(G) for all graphs

gamma_b(G) is at most (5/4) times ir_b(G)

the ratio of IR_b to Gamma_b is unbounded in general graphs

Abstract

A broadcast on a nontrivial connected graph G=(V,E) is a function f from V(G) to {0,1,...,diam(G)} such that f(v) does not exceed the eccentricity of v. The cost of f is the sum of the function values. A broadcast f is dominating if each vertex of G is at distance at most f(v) from a vertex v with positive f(v). We use properties of minimal dominating broadcasts to define the concept of an irredundant broadcast on G. We determine conditions under which an irredundant broadcast is maximal irredundant. Denoting the minimum costs of dominating and maximal irredundant broadcasts by gamma_{b}(G) and ir_{b}(G) respectively, the definitions imply that ir_{b}(G) is bounded above by gamma_{b}(G) for all graphs. We show that gamma_{b} in turn is bounded above by (5/4)ir_{b}(G) for all graphs G. We also briefly consider the upper broadcast number Gamma_{b}(G) and upper irredundant broadcast number IR_{b}(G), and illustrate that the ratio IR_{b} to Gamma_{b} is unbounded for general graphs.