DOWNHILL DOMINATION IN GRAPHS

Authors

Teresa W. Haynes, East Tennessee State UniversityFollow
Stephen T. Hedetniemi, Clemson UniversityFollow
Jessie D. Jamieson, University of Nebraska - LincolnFollow
William B. Jamieson, University of Nebraska - LincolnFollow

Document Type

Article

Date of this Version

2014

Citation

Graph Theory 34 (2014) 603–612

Comments

doi:10.7151/dmgt.1760

Abstract

A path π = (v₁, v₂, . . . , v_k+1) iun 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(vi) 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.