Trees with vertex-edge Roman Domination number twice the domination number minus one
DOI:
https://doi.org/10.22199/issn.0717-6279-2020-06-0084Keywords:
Vertex-edge roman dominating set, Dominating set, Branch duplication in treesAbstract
A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V, E) is a function f : V (G) → {0, 1, 2} such that for each edge e = uv either max{f (u), f (v)} ≠ 0 or there exists a vertex w such that either wu ∈ E or wv ∈ E and f (w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by γ veR(G), is the minimum weight of a ve-RDF G. We characterize trees with vertexedge roman domination number equal to twice domination number minus one.
References
H. A. Ahangar, J. Amjadi, M. Chellali, S. Nazari-Moghaddam, and S. M, heikholeslami, “Trees with double roman domination number twice the domination number plus two”, Iranian journal of science and technology, transactions A: Science volume, vol. 43, pp. 1081–1088, Jun. 2019, doi: 10.1007/s40995-018-0535-7
R. Boutrig, M. Chellali, T. W. Haynes, and S. T. Hedetniemi, “Vertex-edge domination in graphs”, Aequationes mathematicae, vol. 90, no. 2, pp. 355–366, May 2015., doi: 10.1007/s00010-015-0354-2
M. Chellali and N. J. Rad, “Trees with unique Roman dominating functions of minimum weight”, Discrete mathematics, algorithms and applications, vol. 06, no. 03, Art ID 1450038, Jun. 2014., doi: 10.1142/S1793830914500384
E. J. Cockayne, P. A. Dreyer, S. M. Hedetniemi, and S. T. Hedetniemi, “Roman domination in graphs”, Discrete mathematics, vol. 278, no. 1-3, pp. 11–22, Mar. 2004, doi: 10.1016/j.disc.2003.06.004
B. Krishnakumari, Y. B. Venkatakrishnan, and M. Krzywkowski, “Bounds on the vertex–edge domination number of a tree”, Comptes rendus mathematique, vol. 352, no. 5, pp. 363–366, May 2014, doi: 10.1016/j.crma.2014.03.017
J. R. Lewis, S. T. Hedetniemi, T. W. Haynes, and G. H. Fricke, “Vertex-edge domination”, Utilitas mathematica, vol. 81, pp. 193-213, 2010.
J. W. Peters, “Theoretical and algorithmic results on domination and connectivity”, Ph. D. Thesis, Clemson University, 1986. [On line]. Available: https://bit.ly/34lu9SS
P. R. L. Pushpam and S. Padmapriea, “Global Roman domination in graphs”, Discrete applied mathematics, vol. 200, pp. 176–185, Feb. 2016, doi: 10.1016/j.dam.2015.07.014
E. N. Satheesh, “Some variations of domination and applications”, Ph. D. Thesis, Mahatma Gandhi University, 2014. [On line]. Available: https://bit.ly/2HvoErI
X. Zhang, Z. Li, H. Jiang, and Z. Shao, “Double Roman domination in trees”, Information processing letters, vol. 134, pp. 31–34, Jun. 2018, doi: 10.1016/j.ipl.2018.01.004
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Copyright (c) 2020 H. Naresh Kumar, Y. B. Venkatakrishnan
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