The Distance Roman Domination Numbers of Graphs

• Autorzy: Hamideh Aram , Sepideh Norouzian , Seyed Mahmoud Sheikholeslami
• Języki publikacji: 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.

Słowa kluczowe

EN

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

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2013
• Tom: 33
• Numer: 4
• Strony: 717-730

Daty

wydano

2013-09-01

online

2013-10-15

Twórcy

autor

Hamideh Aram

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

autor

Sepideh Norouzian

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

autor

Seyed Mahmoud Sheikholeslami

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

Bibliografia

• [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[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.
• [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[Crossref]
• [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[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).

Identyfikatory

DOI

10.7151/dmgt.1703