Comparing Upper Broadcast Domination and Boundary Independence Numbers of Graphs

TL;DR

This paper compares two graph parameters, the upper broadcast domination number and the boundary independence number, analyzing their relationships, bounds, and differences across various types of graphs.

Contribution

It establishes that neither parameter bounds the other, explores their unbounded differences, and examines ratio bounds in bipartite and general graphs.

Findings

The difference between the parameters can be arbitrarily large.

The ratio of boundary independence to upper broadcast domination is bounded for all graphs.

The ratio of upper broadcast domination to boundary independence is bounded for bipartite graphs but unbounded in general.

Abstract

A broadcast on a nontrivial connected graph G with vertex set V is a function f from V to {0,1,...,diam(G)} such that f(v) is at most the eccentricity of v for all v in V. The weight of f is the sum of the function values taken over V. A vertex u hears f from v if f(v) is positive and d(u,v) is at most f(v). A broadcast f is dominating if every vertex of G hears f. The upper broadcast number of G is {\Gamma}_{b}(G), which is the maximum weight of a minimal dominating broadcast on G. A broadcast f is boundary independent if, for any vertex w that hears f from vertices v_{1},...,v_{k}, where k is at least 2, the distance d(w,v_{i}) equals f(v_{i}) for each i. The maximum weight of a boundary independent broadcast is the boundary independence broadcast number {\alpha}_{bn}(G). We compare {\alpha}_{bn} to {\Gamma}_{b}, showing that neither is an upper bound for the other. We show that the differences {\Gamma}_{b}-{\alpha}_{bn} and {\alpha}_{bn}-{\Gamma}_{b} are unbounded, the ratio {\alpha}_{bn}/{\Gamma}_{b} is bounded for all graphs, and {\Gamma}_{b}/{\alpha}_{bn} is bounded for bipartite graphs but unbounded in general.