On The Co-Roman Domination in Graphs

Open access


Let G = (V, E) be a graph and let f : V (G) → {0, 1, 2} be a function. A vertex v is said to be protected with respect to f, if f(v) > 0 or f(v) = 0 and v is adjacent to a vertex of positive weight. The function f is a co-Roman dominating function if (i) every vertex in V is protected, and (ii) each vV with positive weight has a neighbor uV with f(u) = 0 such that the function fuv : V → {0, 1, 2}, defined by fuv(u) = 1, fuv(v) = f(v) − 1 and fuv(x) = f(x) for xV \ {v, u}, has no unprotected vertex. The weight of f is ω(f) = ∑vV f(v). The co-Roman domination number of a graph G, denoted by γcr(G), is the minimum weight of a co-Roman dominating function on G. In this paper, we give a characterization of graphs of order n for which co-Roman domination number is 2n3 or n − 2, which settles two open problem in [S. Arumugam, K. Ebadi and M. Manrique, Co-Roman domination in graphs, Proc. Indian Acad. Sci. Math. Sci. 125 (2015) 1–10]. Furthermore, we present some sharp bounds on the co-Roman domination number.

[1] H. Abdollahzadeh Ahangar, M.A. Henning, V. Samodivkin and I.G. Yero, Total Roman domination in graphs, Appl. Anal. Discrete Math. 10 (2016) 501–517. doi:10.2298/AADM160802017A

[2] S. Arumugam, K. Ebadi and M. Manrique, Co-Roman dominaton in graphs, Proc. Indian Acad. Sci. Math. Sci. 125 (2015) 1–10. doi:10.1007/s12044-015-0209-8

[3] R.A. Beeler, T.W. Haynes and S.T. Hedetniemi, Double Roman domination, Discrete Appl. Math. 211 (2016) 23–29. doi:10.1016/j.dam.2016.03.017

[4] 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

[5] M. Chellali, T.W. Haynes, S.T. Hedetniemi and A. McRae, Roman {2}-domination, Discrete Appl. Math. 204 (2016) 22–28. doi:10.1016/j.dam.2015.11.013

[6] E.J. Cockayne, P.A. Dreyer Jr., S.M. Hedetniemi and S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004) 11–22. doi:10.1016/j.disc.2003.06.004

[7] O. Favaron, H. Karami, R. Khoeilar and S.M. Sheikholeslami, On the Roman domination number of a graph, Discrete Math. 309 (2009) 3447–3451. doi:10.1016/j.disc.2008.09.043

[8] J. Fink, M. Jacobson, L. Kinch and J. Roberts, On graphs having domination number half their order, Period. Math. Hungar. 16 (1985) 287–293. doi:10.1007/BF01848079

[9] T.W. Haynes and S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dakker Inc., New York, 1998).

[10] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker Inc., New York, 1998).

[11] M.A. Henning and S.T. Hedetniemi, Defending the Roman Empire—A new strategy, Discrete Math. 266 (2003) 239–251. doi:10.1016/S0012-365X(02)00811-7

[12] C. Payan and N.H. Xuong, Domination-balanced graphs, J. Graph Theory 6 (1982) 23–32. doi:10.1002/jgt.3190060104

[13] 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

[14] I. Stewart, Defend the Roman Empire, Sci. Amer. 281 (1999) 136–138. doi:10.1038/scientificamerican1299-136

[15] Z. Zhang, Z. Shao and X. Xu, On the Roman domination numbers of generalized Petersen graphs, J. Combin. Math. Combin. Comput. 89 (2014) 311–320.

Discussiones Mathematicae Graph Theory

The Journal of University of Zielona Góra

Journal Information

IMPACT FACTOR 2017: 0.601
5-year IMPACT FACTOR: 0.535

CiteScore 2017: 0.64

SCImago Journal Rank (SJR) 2017: 0.633
Source Normalized Impact per Paper (SNIP) 2017: 1.095

Mathematical Citation Quotient (MCQ) 2017: 0.36

Target Group

researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 3981 3981 398
PDF Downloads 72 72 15