The Distance Roman Domination Numbers of Graphs
Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Ro...
Gespeichert in:
Veröffentlicht in: | Discussiones Mathematicae. Graph Theory 2013-01, Vol.33 (4), p.717-730 |
---|---|
Hauptverfasser: | , , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Roman dominating function f is the value w(f) =∑
f(v). The k-distance Roman domination number of a graph G, denoted by γ
(D), equals the minimum weight of a k-distance Roman dominating function on G. Note that the 1-distance Roman domination number γ
(G) is the usual Roman domination number
(G). In this paper, we investigate properties of the k-distance Roman domination number. In particular, we prove that for any connected graph G of order n ≥ k +2, γ
(G) ≤ 4n/(2k +3) and we characterize all graphs that achieve this bound. Some of our results extend these ones given by Cockayne et al. in 2004 and Chambers et al. in 2009 for the Roman domination number. |
---|---|
ISSN: | 2083-5892 1234-3099 2083-5892 |
DOI: | 10.7151/dmgt.1703 |