Broadcasts in Graphs: Diametrical Trees
Date
2017-08-21
Authors
Gemmrich, L.
Mynhardt, C.M.
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Abstract
A dominating broadcast on a graph G=(V,E) is a function f:V→{0,1,…,diam(G)} such that f(v)≤e(v) (the eccentricity of v) for all v∈V, and each u∈V is at distance at most f(v) from a vertex v with f(v)≥1. The upper broadcast domination number of G is Γ_{b}(G)=max{∑_{v∈V}f(v):f is a minimal dominating broadcast on G}. As shown by Erwin in [D. Erwin, Cost domination in graphs, Doctoral dissertation, Western Michigan University, 2001], Γ_{b}(G)≥diam(G) for any graph G.
We investigate trees whose upper broadcast domination number equal their diameter and, among more general results, characterise caterpillars with this property.
Description
Accepted for publication in the Australasian Journal of Combinatorics
Keywords
broadcast on a graph, dominating broadcast, minimal dominating broadcast, upper broadcast domination number