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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.38 no.2 Antofagasta jun. 2019

http://dx.doi.org/10.4067/S0716-09172019000200295 

Articles

Total domination and vertex-edge domination in tres

Y. B. Venkatakrishnan1 

H. Naresh Kumar2 

C. Natarajan3 

1 Sastra Deemed University, Department of Mathematics, Tanjore, Tamilnadu, India. e-mail: ybvenkatakrishnan2@gmail.com

2 Sastra Deemed University, Department of Mathematics, Tanjore, Tamilnadu, India. e-mail: nareshhari1403@gmail.com

3 Sastra Deemed University, Department of Mathematics, Tanjore, Tamilnadu, India. e-mail: natarajan@maths.sastra.edu

Abstract:

A vertex v of a graph G = (V,E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a vertex-edge dominating set of G is the vertex-edge domination number γve(G) . In this paper we prove (γt(T)−ℓ+1)/2 ≤ γve(T) ≤(γt(T)+ℓ−1)/2 and characterize trees attaining each of these bounds.

Keywords: Vertex-edge dominating set; total dominating set; trees

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Full text available only in PDF format.

References

[1] R. Boutrig, M. Chellali, T. W. Haynes and S. T. Hedetniemi, Vertex-edge domination in graphs. Aequat. Math., 90, pp. 355-366, (2016). [ Links ]

[2] M. A. Henning and A. Yeo, Total Domination in Graphs (Springer Monographs in Mathematics). 2013. ISBN: 978-1-4614-6524-9 (Print) 978-1-4614-6525-6 (Online). [ Links ]

[3] B. Krishnakumari, Y. B. Venkatakrishnan and M. Krzywkowski, Bounds on the vertex-edge domination number of a tree. C. R. Acad. Sci. Paris, Ser.I 352, pp. 363-366, (2014). [ Links ]

[4] J. R. Lewis, S. T. Hedetniemi, T. W. Haynes and G. H. Fricke, Vertex-edge domination. Util. Math. 81, pp. 193-213, (2010). [ Links ]

[5] J. W. Peters. Theoretical and algorithmic results on domination and connectivity. Ph.D. Thesis, Clemson University, (1986) [ Links ]

Received: July 2017; Accepted: January 2019

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License