## Mathematics, Department of

#### Date of this Version

2014

#### Citation

Graph Theory 34 (2014) 603–612

#### Abstract

A path π = (v_{1}, v_{2}, . . . , 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.

## Comments

doi:10.7151/dmgt.1760