Downhill domination in graphs

A path $π = (v_1, v_2, . . ., v_{k+1})$ in a graph $G = (V,E)$ is a downhill path if for every $i, 1 ≤ i ≤ k, deg(v_i) ≥ deg(v_{i+1})$, where $deg(v_i)$ denotes the degree of vertex $v_i ∈ V$. 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.