DMGT

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

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 ∈ V}f (v). The k-distance Roman domination number of a graph G, denoted by γ^k_R(D), equals the minimum weight of a k-distance Roman dominating function on G. Note that the 1-distance Roman domination number γ¹_R(G) is the usual Roman domination number γ_R(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_R^k(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.

References

[1]	J.A. Bondy, U.S.R. Murty, Graph Theory with Applications (The Macmillan Press Ltd. London and Basingstoke, 1976).
[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.
[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.
[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.
[5]	O. Favaron, H. Karami and S.M. Sheikholeslami, On the Roman domination number in graphs, Discrete Math. 309 (2009) 3447--3451, doi: 10.1016/j.disc.2008.09.043.
[6]	T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc. NewYork, 1998).
[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.
[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).

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