On The Roman Domination Stable Graphs

Open access


A Roman dominating function (or just RDF) on a graph G = (V,E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF f is the value f(V (G)) = Pu2V (G) f(u). The Roman domination number of a graph G, denoted by R(G), is the minimum weight of a Roman dominating function on G. A graph G is Roman domination stable if the Roman domination number of G remains unchanged under removal of any vertex. In this paper we present upper bounds for the Roman domination number in the class of Roman domination stable graphs, improving bounds posed in [V. Samodivkin, Roman domination in graphs: the class RUV R, Discrete Math. Algorithms Appl. 8 (2016) 1650049].

[1] E.W. Chambers, W.B. Kinnersley, N. Prince and D.B. West, Extremal Problems for Roman Domination, SIAM J. Discrete Math. 23 (2009) 1575-1586. doi:

[2] M. Chellali and N. Jafari Rad, Trees with unique Roman dominating functions of minimum weight, Discrete Math. Algorithms Appl. 6 (2014) 1450038. doi:

[3] 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:

[4] A. Hansberg, N. Jafari Rad and L. Volkmann, Vertex and edge critical Roman domination in graphs, Util. Math. 92 (2013) 73-88.

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

[6] N. Jafari Rad and L. Volkmann, Changing and unchanging the Roman domination number of a graph, Util. Math. 89 (2012) 79-95.

[7] C.-H. Liu and G.J. Chang, Roman domination on strongly chordal graphs, J. Comb. Optim. 26 (2013) 608-619. doi:

[8] C.-H. Liu and G.J. Chang, Upper bounds on Roman domination numbers of graphs, Discrete Math. 312 (2012) 1386-1391. doi:

[9] V. Samodivkin, On the Roman bondage number of graphs on surfaces, Int. J. Graph Theory Appl. 1 (2015) 67-75. doi:

[10] V. Samodivkin, Roman domination in graphs: the class RUV R, Discrete Math. Algorithms Appl. 8 (2016) 1650049. doi:

[11] P.J. Slater, Dominating and reference sets in graphs, J. Math. Phys. Sci. 22 (1988) 445-455.

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 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.763
Source Normalized Impact per Paper (SNIP) 2018: 0.934

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 211 193 33
PDF Downloads 110 101 13