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• # Artykuł - szczegóły

## Discussiones Mathematicae Graph Theory

2013 | 33 | 4 | 717-730

## The Distance Roman Domination Numbers of Graphs

EN

### Abstrakty

EN
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) =∑v∈V f(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, γkR (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.

EN

717-730

wydano
2013-09-01
online
2013-10-15

### Twórcy

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

### Bibliografia

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• [2] E.W. Chambers, B. Kinnersley, N. Prince and D.B. West, Extremal problems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575-1586. doi:10.1137/070699688[WoS][Crossref]
• [3] E.J. Cockayne, P.M. Dreyer Jr., S.M. Hedetniemi and S.T. Hedetniemi, On Roman domination in graphs, Discrete Math. 278 (2004) 11-22. doi:10.1016/j.disc.2003.06.004[Crossref]
• [4] E.J. Cockayne, P.J.P. Grobler, W.R. Gründlingh, J. Munganga, and J.H. van Vuuren, Protection of a graph, Util. Math. 67 (2005) 19-32.
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• [7] B.P. Mobaraky and S.M. Sheikholeslami, Bounds on Roman domination numbers of a graph, Mat. Vesnik 60 (2008) 247-253.
• [8] C.S. ReVelle and K.E. Rosing, Defendens imperium romanum: a classical problem in military strategy, Amer. Math. Monthly 107 (2000) 585-594. doi:10.2307/2589113[Crossref]
• [9] I. Stewart, Defend the Roman Empire, Sci. Amer. 281 (1999) 136-139.
• [10] D.B. West, Introduction to Graph Theory (Prentice-Hall, Inc, 2000).