# Downhill Domination in Graphs

• Volume: 34, Issue: 3, page 603-612
• ISSN: 2083-5892

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## Abstract

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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

## How to cite

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Teresa W. Haynes, et al. "Downhill Domination in Graphs." Discussiones Mathematicae Graph Theory 34.3 (2014): 603-612. <http://eudml.org/doc/267807>.

@article{TeresaW2014,
abstract = {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},
author = {Teresa W. Haynes, Stephen T. Hedetniemi, Jessie D. Jamieson, William B. Jamieson},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {downhill path; downhill domination number},
language = {eng},
number = {3},
pages = {603-612},
title = {Downhill Domination in Graphs},
url = {http://eudml.org/doc/267807},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Teresa W. Haynes
AU - Stephen T. Hedetniemi
AU - Jessie D. Jamieson
AU - William B. Jamieson
TI - Downhill Domination in Graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2014
VL - 34
IS - 3
SP - 603
EP - 612
AB - 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
LA - eng
KW - downhill path; downhill domination number
UR - http://eudml.org/doc/267807
ER -

## References

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1. [1] T.W. Haynes, S.T. Hedetniemi, J. Jamieson and W. Jamieson, Downhill and uphill domination in graphs, submitted for publication (2013). Zbl1295.05176
2. [2] P. Hall, On representation of subsets, J. London Math. Soc. 10 (1935) 26-30. Zbl0010.34503
3. [3] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, 1998). Zbl0890.05002
4. [4] J.D. Hedetniemi, S.M. Hedetniemi, S.T. Hedetniemi and T. Lewis, Analyzing graphs by degrees, AKCE Int. J. Graphs Comb., to appear. Zbl1293.05051
5. [5] O. Ore, Theory of Graphs (Amer. Math. Soc. Colloq. Publ. 38, 1962).

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