On The Total Roman Domination in Trees

Open access

Abstract

A total Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the following conditions: (i) every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2 and (ii) the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dominating function f is the value f(V (G)) = ΣuV(G)f (u). The total Roman domination number γtR(G) is the minimum weight of a total Roman dominating function of G. Ahangar et al. in [H.A. Ahangar, M.A. Henning, V. Samodivkin and I.G. Yero, Total Roman domination in graphs, Appl. Anal. Discrete Math. 10 (2016) 501–517] recently showed that for any graph G without isolated vertices, 2γ(G) ≤ γtR(G) ≤ 3γ(G), where γ(G) is the domination number of G, and they raised the problem of characterizing the graphs G achieving these upper and lower bounds. In this paper, we provide a constructive characterization of these trees.

[1] H. Abdollahzadeh Ahangar, J. Amjadi, S.M. Sheikholeslami and M. Soroudi, Bounds on the total Roman domination numbers, Ars Combin., to appear.

[2] H. Abdollahzadeh Ahangar, A. Bahremandpour, S.M. Sheikholeslami, N.D. Soner, Z. Tahmasbzadehbaee and L. Volkmann, Maximal Roman domination numbers in graphs, Util. Math. 103 (2017) 245–258.

[3] 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

[4] H. Abdollahzadeh Ahangar, T.W. Haynes and J.C. Valenzuela-Tripodoro, Mixed Roman domination in graphs, Bull. Malays. Math. Sci. Soc. 40 (2017) 1443–1454. doi:10.1007/s40840-015-0141-1

[5] J. Amjadi, S. Nazari-Moghaddam and S.M. Sheikholeslami, Global total Roman domination in graphs, Discrete Math. Algorithms Appl. 09 (2017) ID: 1750050. doi:10.1142/S1793830917500501

[6] J. Amjadi, S. Nazari-Moghaddam, S.M. Sheikholeslami and L. Volkmann, Total Roman domination number of trees, Australas. J. Combin. 69 (2017) 271–285.

[7] J. Amjadi, S.M. Sheikholeslami and M. Soroudi, Nordhaus-Gaddum bounds for total Roman domination, J. Comb. Optim. 35 (2018) 126–133. doi:10.1007/s10878-017-0158-5

[8] 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

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

[10] E.J. Cockayne, P.A. Dreyer, 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

[11] M.R. Garey and D.S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness (W.H. Freeman and Co., San Francisco, Calif., 1979).

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

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

[14] 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

[15] L.L. Kelleher and M.B. Cozzens, Dominating sets in social network graphs, Math. Social Sci. 16 (1988) 267–279. doi:10.1016/0165-4896(88)90041-8

[16] C.-H. Liu and G.J. Chang, Roman domination on strongly chordal graphs, J. Comb. Optim. 26 (2013) 608–619. doi:10.1007/s10878-012-9482-y

[17] C.S. ReVelle and K.E. Rosing, Defendens imperium Romanum: a classical problem in military strategy, Amer. Math. Monthly 107 (2000) 585–594. doi:10.1080/00029890.2000.12005243

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

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

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 3531 3531 351
PDF Downloads 89 89 17