Discussiones Mathematicae Graph Theory 33(4) (2013)
717-730

doi: 10.7151/dmgt.1703

Hamideh Aram, Sepideh Norouzian, Seyed Mahmoud Sheikholeslami
Department of Mathematics | Lutz Volkmann
Lehrstuhl II für Mathematik |

**Keywords:** *k*-distance Roman dominating function, *k*-distance Roman domination number, Roman dominating function, Roman domination number

**2010 Mathematics Subject Classification:** 05C69.

[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