General Bounds on the Downhill Domination Number in Graphs.

• Main Author: Jamieson, William
• Format: Others
• Published: Digital Commons @ East Tennessee State University 2013
• Subjects: Graph Theory; Domination; Degree; Degree Constraints; Mathematics; Physical Sciences and Mathematics
• Online Access: https://dc.etsu.edu/honors/107; https://dc.etsu.edu/cgi/viewcontent.cgi?article=1111&context=honors

A path π = (v1, v2,...vk+1) in a graph G = (V,E) is a downhill path if for every i, 1 < i < k, deg(vi) > 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.