Discussiones Mathematicae Graph Theory 33(4) (2013) 717-730
doi: 10.7151/dmgt.1703

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The distance Roman domination numbers of graphs

Hamideh Aram, Sepideh Norouzian,
Seyed Mahmoud Sheikholeslami

Department of Mathematics
Azarbaijan Shahid Madani University
Tabriz, I.R. Iran

Lutz Volkmann

Lehrstuhl II für Mathematik
RWTH Aachen University
52056 Aachen, Germany


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 ω(f) = ∑v ∈ Vf (v). The k-distance Roman domination number of a graph G, denoted by γkR(D), equals the minimum weight of a k-distance Roman dominating function on G. Note that the 1-distance Roman domination number γ1R(G) is the usual Roman domination number γR(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, γRk(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.

Keywords: k-distance Roman dominating function, k-distance Roman domination number, Roman dominating function, Roman domination number

2010 Mathematics Subject Classification: 05C69.


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Received 20 December 2011
Revised 4 September 2012
Accepted 5 September 2012