2016/10/17 by Erik Insko, Insko, Erik, Bethany Kubik +3
Mathematics · #05C05 #05C69 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C05 #msc:05C69
paper · pdf · doi:10.48550/arxiv.1610.05250
19 pages, 7 figures
arxiv created 2016/10/17 · arxiv updated 2016/10/18
A broadcast on a graph G=(V,E) is a function f:V → \0,1, …, diam(G)\ satisfying f(v) ≤ e(v) for all v ∈ V, where e(v) denotes the eccentricity of v and diam(G) denotes the diameter of G. We say that a broadcast dominates G if every vertex can hear at least one broadcasting node. The upper domination number is the maximum cost of all possible minimal broadcasts, where the cost of a broadcast is defined as cost (f)= ∑v ∈ Vf(v). In this paper we establish both the upper domination number and the upper broadcast domination number on toroidal grids. In addition, we classify all diametrical trees, that is, trees whose upper domination number is equal to its diameter.