Original Research Article

Article volume = 2021 and issue = 2

Pages: 97–103

Article publication Date: November, 1, 2021

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# On The Edge Double Roman Domination Number of Planar Graph

#### Mina Valinavaz

Department of Mathematics, Azarbaijan Shahid Madani University Tabriz, I.R. Iran.

##### Abstract:

An edge double Roman dominating function (EDRDF) on a graph G is a function f : E(G) ! f0, 1, 2, 3g satisfying the condition that such that every edge e with f(e) = 0, is adjacent to at least two edge e, e0 for which f(e) = f(e0) = 2 or one edge e00 with f(e00) = 3, and if f(e) = 1, then edge e must have at least one neighbor e0 with f(e0) > 2. The Edge double Roman dominating number of G, denoted by 0 dR(G), is the minimum weight w(f) = P e2E(G) f(e) of an edge double Roman dominating function f of G. In this paper, we introduction some results on the edge double Roman domination number of a graph. Also, we provide some upper and lower bounds for the edge double Roman domination number of graphs.

##### Keywords:

Double Roman dominating function, Double Roman domination number, Edge double Roman dominating function, Edge double Roman domination number.

##### References:

- [1] H. Abdollahzadeh Ahangar, M. Chellai, S.M. Sheikholeslam, em on the double Roman domination in graphs, Appl. Anal. Discrete Math. 10 (2017), 501–517.
- [2] H. Abdollahzadeh Ahangar, H. Jahani and N. Jafary Rad, Rainbow edge domination numbers in graphs,
- [3] S. Akbari, S. Ehsani, S. Ghajar, P. Jalaly Khalilabadi, S. Sadeghian Sadeghabad, On the edge Roman domination in graphs,(manuscript).
- [4] S. Akbari, S. Qajar, On the edge Roman domination number of planar graphs, (manuscript).
- [5] J. Amjadi, S. Nazari-Moghaddam, S.M. Sheikholeslami, and L. Volkmann, An upper bound on the double Roman domination number, Combin. Optim. 36(2018), 81-89.
- [6] S. Arumugam and S. Velammal, Edge domination in graphs, Taiwanese J. Math. 2(1998), 173-179.
- [7] R.A. Beeler, T.W. Haynes, and S.T. Hedetniemi, Double Roman Domination, Discrete Apple. Math. 211(2016), 23-29. 1
- [8] G.J. Chang, S.H. Chen, C.H. Liu, Edge Roman domination on graphs. Graphs. Combin. 32(5)(2016), 1731-1747
- [9] K. Ebadi, E. Khodadadi and L. Pushpalatha, On the Roman edge domination number of a graph, Int. J. Math. Combin. 4(2010), 38-45.
- [10] N. Jafari Rad, A note on the edge Roman domination in trees, Electronic J. Graph Theory and Apple. 5(2017), 1-6.
- [11] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fandamentals of Domination in Graphs, (Marcel DeKKer, New York, (1998).
- [12] S.R. Jararram, Line domination in graphs, Graphs Combin. 3 (1987), 357-363.
- [13] S. Mitchcal and S.T. Hedetniemi, Edge domination in trees, Congr. Numer. 19(1977), 489-509.
- [14] M.N. Paspasan and S.R. Canoy, Jr, Restrainted total edge domination in graphs, Apple. Math. Sci. 9(2015), 7139-7148.
- [15] P. Roushini Leely Pushpam, T.N.M. Malini Mai, Edge Roman domination in graphs, J. Combin. Math. Combin. Comput. 69(2009)175-182.
- [16] D.B. West, Introduction to Graph Theory(Prentice-Hall, Inc, 2000). 3.1, 3.5
- [17] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl., 20 (1967),197–228.
- [18] Y. Yao, Y. J. Cho, Y. C. Liou, R. P. Agarwal, Constructed nets with perturbations for equilibrium and fixed point problems, J. Inequal. Appl., 2014 (2014), 14 pages.
- [19] B. O’Neill, Semi-Riemannian geomerty with applications to relativity, Academic Press, London, (1983).

##### Cite this article as:

- Mina Valinavaz, On The Edge Double Roman Domination Number of Planar Graph, Communications in Combinatorics, Cryptography & Computer Science, 2021(2), PP.97–103, 2021
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