Roman Domination in Graphs

This chapter is concerned with the concept Roman domination in graphs, which was introduced in 2004 by Cockayne, Dreyer, S.M. Hedetniemi, and S.T. Hedetniemi based on the strategies for defending the Roman Empire presented by Stewart (Sci Am 281:136–139, 1999) and ReVelle and Rosing (ReVelle CS, Rosing KE, Am Math Mon 107:585–594, 2000). Since then, more than 180 papers have been published on this topic, where several new variations were defined and that will be exposed in another chapter. In this chapter we will only survey the results on the Roman domination number as well as those when the structure of the graph is modified by the addition of edges/vertices or removing edges/vertices.

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[2]  V. Samodivkin.  Roman domination excellent graphs: trees , 2016, 1610.00297.

[3]  Teresa W. Haynes,et al.  LOWER BOUNDS ON THE ROMAN AND INDEPENDENT ROMAN DOMINATION NUMBERS , 2016 .

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[6]  Yunjian Wu.  An improvement on Vizing's conjecture , 2013, Inf. Process. Lett..

[7]  Erin W. Chambers,et al.  Extremal Problems for Roman Domination , 2009, SIAM J. Discret. Math..

[8]  Ivona Puljić,et al.  Some results for Roman domination number on Cardinal product of paths and cycles , 2014 .

[9]  Martin Farber,et al.  Domination, independent domination, and duality in strongly chordal graphs , 1984, Discret. Appl. Math..

[10]  Xin Chen,et al.  A note on Roman domination in graphs , 2006, Discret. Math..

[11]  Ismael G. Yero,et al.  Roman domination in Cartesian product graphs and strong product graphs , 2011, 1111.3517.

[12]  Gerard J. Chang,et al.  Roman domination on strongly chordal graphs , 2013, J. Comb. Optim..

[13]  Seyed Mahmoud Sheikholeslami,et al.  On the Roman bondage Number of a Graph , 2011, Discret. Math. Algorithms Appl..

[14]  Mostafa Blidia,et al.  Criticality indices of Roman domination of paths and cycles , 2013, Australas. J Comb..

[15]  Michael A. Henning.  A characterization of Roman trees , 2002, Discuss. Math. Graph Theory.

[16]  Ermelinda DeLaViña,et al.  On Total Domination in Graphs , 2012 .

[18]  Seyed Mahmoud Sheikholeslami,et al.  Roman domination subdivision number of a graph and its complement , 2013, Ars Comb..

[19]  Upper bounds for the domination subdivision and bondage numbers of graphs on topological surfaces , 2013 .

[20]  Jun-Ming Xu,et al.  Roman bondage numbers of some graphs , 2014, Australas. J Comb..

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[23]  Vadim E. Zverovich,et al.  On Roman, Global and Restrained Domination in Graphs , 2011, Graphs Comb..

[24]  Simone Dantas,et al.  Averaging 2-rainbow domination and Roman domination , 2016, Discret. Appl. Math..

[25]  E. Cockayne,et al.  Properties of Hereditary Hypergraphs and Middle Graphs , 1978, Canadian Mathematical Bulletin.

[26]  Vladimir Samodivkin.  Upper bounds for domination related parameters in graphs on surfaces , 2016, AKCE Int. J. Graphs Comb..

[27]  Lutz Volkmann,et al.  Roman bondage in graphs , 2011, Discuss. Math. Graph Theory.

[28]  Chun-Hung Liu,et al.  Upper bounds on Roman domination numbers of graphs , 2012, Discret. Math..

[29]  Hua-ming Xing,et al.  Note on 2-rainbow domination and Roman domination in graphs , 2010, Appl. Math. Lett..

[30]  Saieed Akbari,et al.  A Note on the Roman Bondage Number of Planar Graphs , 2013, Graphs Comb..

[31]  Seyed Mahmoud Sheikholeslami,et al.  On the Roman domination number of a graph , 2009, Discret. Math..

[32]  Nader Jafari Rad,et al.  On The Roman Domination Stable Graphs , 2017, Discuss. Math. Graph Theory.

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[34]  Michael A. Henning,et al.  RAINBOW DOMINATION IN GRAPHS , 2008 .

[35]  S. M. Sheikholeslami,et al.  Roman game domination subdivision number of a graph , 2013 .

[36]  Mathieu Liedloff,et al.  Efficient algorithms for Roman domination on some classes of graphs , 2008, Discret. Appl. Math..

[37]  Antoaneta Klobučar,et al.  Roman domination number on cardinal product of paths and cycles , 2015 .

[38]  Yang Yuansheng,et al.  Roman domination in regular graphs , 2009 .

[39]  Chun-Hung Liu,et al.  Roman Domination on 2-Connected Graphs , 2012, SIAM J. Discret. Math..

[40]  Lutz Volkmann,et al.  Roman Domination Dot-critical Graphs , 2013, Graphs Comb..

[41]  Sergio Bermudo.  On the differential and Roman domination number of a graph with minimum degree two , 2017, Discret. Appl. Math..

[42]  Seyed Mahmoud Sheikholeslami,et al.  The Roman domatic number of a graph , 2010, Appl. Math. Lett..

[43]  Frank Harary,et al.  Domination alteration sets in graphs , 1983, Discret. Math..

[44]  Hongyu Liang,et al.  Computing Roman domatic number of graphs , 2016, Inf. Process. Lett..

[45]  Tadeja Kraner Sumenjak,et al.  On the Roman domination in the lexicographic product of graphs , 2012, Discret. Appl. Math..

[46]  Nader Jafari Rad.  Roman domination edge critical graphs having precisely two cycles , 2017, Ars Comb..

[47]  Saieed Akbari,et al.  A Note on Roman Bondage Number of Graphs , 2016, Ars Comb..

[48]  Stephen T. Hedetniemi,et al.  Roman domination in graphs , 2004, Discret. Math..

[49]  Critical properties on Roman domination graphs , 2013, 1311.4476.

[50]  Vladimir Samodivkin.  Roman domination in graphs: The class ℛUV R , 2016, Discret. Math. Algorithms Appl..

[51]  Nader Jafari Rad,et al.  A Nordhaus-Gaddum bound for Roman domination , 2019, Discret. Math. Algorithms Appl..

[52]  Mostafa Blidia,et al.  Extremal Graphs for a Bound on the Roman Domination Number , 2020, Discuss. Math. Graph Theory.

[53]  Jun-Ming Xu,et al.  On Bondage Numbers of Graphs: A Survey with Some Comments , 2012, 1204.4010.

[54]  Polona Pavlic,et al.  Roman Domination Number of the Cartesian Products of Paths and Cycles , 2012, Electron. J. Comb..

[55]  Vladimir Samodivkin.  On the Roman Bondage Number of Graphs on surfaces , 2014 .

[56]  Lutz Volkmann,et al.  On the Roman Bondage Number of Planar Graphs , 2011, Graphs Comb..

[57]  Lutz Volkmann,et al.  Changing and unchanging the Roman domination number of a graph , 2012 .

[58]  Peter J. Slater,et al.  Fundamentals of domination in graphs , 1998, Pure and applied mathematics.

[59]  Shinya Fujita,et al.  Difference between 2-rainbow domination and Roman domination in graphs , 2013, Discret. Appl. Math..

[60]  Nader Jafari Rad,et al.  On 2-rainbow domination and Roman domination in graphs , 2013, Australas. J Comb..

[61]  Lutz Volkmann,et al.  Vertex and edge critical Roman domination in graphs , 2013 .

[62]  Teresa W. Haynes,et al.  A Roman Domination Chain , 2016, Graphs Comb..

[63]  Nader Jafari Rad,et al.  Trees with unique Roman dominating functions of minimum weight , 2014, Discret. Math. Algorithms Appl..

[64]  Seyed Mahmoud Sheikholeslami,et al.  Roman domination subdivision number of graphs , 2009 .

[65]  Seyed Mahmoud Sheikholeslami,et al.  Total Roman Reinforcement in Graphs , 2019, Discuss. Math. Graph Theory.