# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## 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

## Abstract

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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