A class of trees with equal broadcast and domination numbers

A broadcast on a graph G is a function f : V → {0, 1, 2, . . . }. The broadcast number of G is the minimum value of ∑ v∈V f(v) among all broadcasts f for which each vertex of G is within distance f(v) from some vertex v with f(v) ≥ 1. The broadcast number is bounded above by the radius and the domination number of G. We consider a class of trees that contains the caterpillars and characterize the trees in this class that have equal domination and broadcast numbers, thus generalizing the results in: [S.M. Seager, Dominating broadcasts of caterpillars, Ars Combin. 88 (2008), 307–319].