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Every graph G contains a minimum vertex-coloring with the property that at least one color class of the coloring is a maximal independent set (equivalently, a dominating set) in G. Among all such minimum vertex-colorings of the vertices of G, a coloring with the maximum number of color classes that are dominating sets in G is called a dominating-χ-coloring of G. The number of color classes that are...
A broadcast on a graph G is a function f:V→Z+∪{0}. The broadcast number of G is the minimum value of ∑v∈Vf(v) among all broadcasts f for which each vertex of G is within distance f(v) from some vertex v with f(v)≥1. This number is bounded above by the radius and the domination number of G. We show that to characterize trees with equal broadcast and domination numbers it is sufficient to characterize...
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