Bounds on signed total double Roman domination

Document Type : Original paper

Authors

1 Azarbaijan Shahid Madani University

2 Babol Noshirvani University of Technology

Abstract

A signed total double Roman dominating function (STDRDF) on {an} isolated-free graph G=(V,E) is a function f:V(G)→{−1,1,2,3} such that (i) every vertex v with f(v)=−1 has at least two neighbors assigned 2 under f or one neighbor w with f(w)=3, (ii) every vertex v with f(v)=1 has at least one neighbor w with f(w)≥2 and (iii) ∑u∈N(v)f(u)≥1 holds for any vertex v. The weight of {an} STDRDF is the value f(V(G))=∑u∈V(G)f(u). The signed total double Roman domination number γsdRt(G) is the minimum weight of an STDRDF on G. In this paper, we continue the study of the signed total double Roman domination in graphs and present some sharp bounds for this parameter.

Keywords

Main Subjects

References

[1] H. Abdollahzadeh Ahangar, J. Amjadi, M. Atapour, M. Chellali, and S.M. Sheikholeslami, Double Roman trees, Ars Combin. 145 (2019), 173–183.
[2] H. Abdollahzadeh Ahangar, J. Amjadi, M. Chellali, S. Nazari-Moghaddam, and S.M. Sheikholeslami, Trees with double Roman domination number twice the domination number plus two, Iran. J. Sci. Technol. Trans. A, Sci. 43 (2019), no. 3,
1081–1088.
[3] H. Abdollahzadeh Ahangar, J. Amjadi, S.M. Sheikholeslami, L. Volkmann, and Y. Zhao, Signed Roman edge domination numbers in graphs, J. Comb. Optim. 31 (2016), no. 1, 333–346.
[4] H. Abdollahzadeh Ahangar, L. Asgharsharghi, S.M. Sheikholeslami, and L. Volkmann, Signed mixed Roman domination numbers in graphs, J. Comb. Optim. 32 (2016), no. 1, 299–317.
[5] H. Abdollahzadeh Ahangar, M. Chellali, and S.M. Sheikholeslami, On the double Roman domination in graphs, Discrete Appl. Math. 232 (2017), 1–7.
[6] H. Abdollahzadeh Ahangar, M. Chellali, and S.M. Sheikholeslami, Signed double Roman domination in graphs, Discrete Appl. Math. 257 (2019), 1–11.
[7] H. Abdollahzadeh Ahangar, M. Chellali, and S.M. Sheikholeslami, Signed double Roman domination in graphs, Filomat 33 (2019), no. 1, 121–134.
[8] H. Abdollahzadeh Ahangar, M.A. Henning, C. Löwenstein, Y. Zhao, and V. Samodivkin, Signed Roman domination in graphs, J. Comb. Optim. 27 (2014), no. 2, 241–255.
[9] H. Abdollahzadeh Ahangar, R. Khoeilar, L. Shahbazi, and S.M. Sheikholeslami, Signed total double Roman domination, Ars Combin. (to appea).
[10] H. Amjadi, J.and Yang, S. Nazari-Moghaddam, S.M. Sheikholeslami, and Z. Shao, Signed double Roman k-domination in graphs., Australas. J Comb. 72 (2018), 82–105.
[11] R.A. Beeler, T.W. Haynes, and S.T. Hedetniemi, Double Roman domination, Discrete Appl. Math. 211 (2016), 23–29.
[12] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, A survey on Roman domination parameters in directed graphs, J. Combin. Math. Combin. Comput. (to appear).
[13] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, The Roman domatic problem in graphs and digraphs: A survey, Discuss. Math. Graph Theory (in press).
[14] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Varieties of Roman domination II, AKCE Int. J. Graphs Comb. (in
press).
[15] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Structures of Domination in Graphs, (eds), T.W. haynes, S.T. hedetniemi and M.A. henning, ch. Varieties of Roman domination, Springer, 2021.
[16] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Topics in Domination in Graphs, (eds), T.W. haynes, S.T. hedetniemi and M.A. henning, ch. Roman domination in graphs, Springer, 2020.
[17] E.J. Cockayne, P.A. Dreyer Jr, S.M. Hedetniemi, and S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004), no. 1-3, 11–22.
[18] C.S. ReVelle and K.E. Rosing, Defendens imperium romanum: a classical problem in military strategy, Amer. Math. Monthly 107 (2000), no. 7, 585–594.
[19] I. Stewart, Defend the Roman empire!, Sci. Amer. 281 (1999), no. 6, 136–138.
[20] D.B. West, Introduction to Graph Theory, Prentice Hall, USA, 2001.
[21] H. Yang, P. Wu, S. Nazari-Moghaddam, S.M. Sheikholeslami, X. Zhang, Z. Shao, and Y.Y. Tang, Bounds for signed double Roman k-domination in trees, RAIRO, Oper. Res. 53 (2019), no. 2, 627–643.
 
Communications in Combinatorics and Optimization
Volume 5, Issue 2
December 2020
Pages 191-206

Files

Share

How to cite

Statistics