General Bounds on the Downhill Domination Number in Graphs.

A path π = (v1, v2,...vk+1) in a graph G = (V,E) is a downhill path if for every i, 1 \u3c i \u3c k, deg(vi) \u3e deg(vi+1), where deg(vi) denotes the degree of vertex vi ∊ V. The downhill domination number equals the minimum cardinality of a set S ⊂ V having the property that every vertex v ∊ V lies on a downhill path originating from some vertex in S. We investigate downhill domination numbers of graphs and give upper bounds. In particular, we show that the downhill domination number of a graph is at most half its order, and that the downhill domination number of a tree is at most one third its order. We characterize the graphs obtaining each of these bounds