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New bounds for the broadcast domination number of a graph

  • Research Article
  • Published:
Central European Journal of Mathematics

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 such that each vertex is within distance f(v) from a vertex v with f(v) > 0. The cost of a broadcast f is σ(f) = Σ v∈V f(v), and the broadcast number λ b (G) is the minimum cost of a dominating broadcast.

A set X ⊆ V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir (G) is the cardinality of a smallest maximal irredundant set of G.

We prove the bound λb(G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir (G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for λb.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Thompson Rivers University, 900 McGill Road, Kamloops, BC, V2C 0C8, Canada

    Richard C. Brewster

  2. Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, Victoria, BC, V8W 3P4, Canada

    Christina M. Mynhardt & Laura E. Teshima

Authors
  1. Richard C. Brewster
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  2. Christina M. Mynhardt
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  3. Laura E. Teshima
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Correspondence to Richard C. Brewster.

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Brewster, R.C., Mynhardt, C.M. & Teshima, L.E. New bounds for the broadcast domination number of a graph. centr.eur.j.math. 11, 1334–1343 (2013). https://doi.org/10.2478/s11533-013-0234-8

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  • DOI: https://doi.org/10.2478/s11533-013-0234-8

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