Upper Broadcast Domination Number of Caterpillars with no Trunks

Bouchouika, Sabrina (University of Sciences and Technology Houari Boumediene (USTHB))
Bouchemakh, Isma (University of Sciences and Technology Houari Boumediene (USTHB))
Sopena, Eric (Bordeaux University, CNRS, Bordeaux INP, LaBRI.)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 12, No 2 (2024): Electronic Journal of Graph Theory and Applications

A broadcast on a graph G = (V,E) is a function f : V →{0,…,diam(G)} such that f(v) ≤ eG(v) for every vertex v ∈ V , where diam(G) denotes the diameter of G and eG(v) the eccentricity of v in G. Such a broadcast f is minimal if there does not exist any broadcast g≠f on G such that g(v) ≤ f(v) for all v ∈ V . The upper broadcast domination number of G is the maximum value of ∑ v∈V f(v) among all minimal broadcasts f on G for which each vertex of G is at distance at most f(v) from some vertex v with f(v) ≥ 1. In this paper, we study the minimal dominating broadcasts of caterpillars and give the exact value of the upper broadcast domination number of caterpillars with no trunks.