# The Distance Roman Domination Numbers of Graphs

Hamideh Aram; Sepideh Norouzian; Seyed Mahmoud Sheikholeslami

Discussiones Mathematicae Graph Theory (2013)

- Volume: 33, Issue: 4, page 717-730
- ISSN: 2083-5892

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topHamideh Aram, Sepideh Norouzian, and Seyed Mahmoud Sheikholeslami. "The Distance Roman Domination Numbers of Graphs." Discussiones Mathematicae Graph Theory 33.4 (2013): 717-730. <http://eudml.org/doc/268287>.

@article{HamidehAram2013,

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 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.},

author = {Hamideh Aram, Sepideh Norouzian, Seyed Mahmoud Sheikholeslami},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {k-distance Roman dominating function; k-distance Roman domination number; Roman dominating function; Roman domination number; -distance Roman dominating function; -distance Roman domination number},

language = {eng},

number = {4},

pages = {717-730},

title = {The Distance Roman Domination Numbers of Graphs},

url = {http://eudml.org/doc/268287},

volume = {33},

year = {2013},

}

TY - JOUR

AU - Hamideh Aram

AU - Sepideh Norouzian

AU - Seyed Mahmoud Sheikholeslami

TI - The Distance Roman Domination Numbers of Graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2013

VL - 33

IS - 4

SP - 717

EP - 730

AB - 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.

LA - eng

KW - k-distance Roman dominating function; k-distance Roman domination number; Roman dominating function; Roman domination number; -distance Roman dominating function; -distance Roman domination number

UR - http://eudml.org/doc/268287

ER -

## References

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- [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] Zbl1036.05034
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- [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc. NewYork, 1998). Zbl0890.05002
- [7] B.P. Mobaraky and S.M. Sheikholeslami, Bounds on Roman domination numbers of a graph, Mat. Vesnik 60 (2008) 247-253. Zbl1274.05359
- [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] Zbl1039.90038
- [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).

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