ON TOTAL VERTEX-EDGE DOMINATION

Abstract

In this paper we obtain an improved upper bound of total vertex edgem domination number of a tree. If T is a connected tree with order n, then γtve T ≤n with m = 6de and we characterize the trees attaining this upper bound. Furthermorevewe provide a characterization of trees T with γt T = γt T

Keywords

• Domination,vertex-edge domination,total vertex-edge domination,total domination.

References

1. [1] Boutrig, R. and Chellali, M., (in press), Total vertex-edge domination, International Journal of Computer Mathematics.
2. [2] Boutrig, R., Chellali, M., Haynes, T. W. and Hedetniemi, S. T., (2016), Vertex-edge domination in graphs, Aequationes Mathematicae, 90(2), pp. 355-366.
3. [3] Haynes, T. W., Hedetniemi, S. T. and Slater, P. J., (1998), Fundamentals of Domination in Graphs, Marcel Dekker, New York.
4. [4] Kang, C. X., (2014), Total domination value in graphs, Util. Math., 95, pp. 263-279.
5. [5] Krishnakumari, B., Venkatakrishnan, Y. B. and Krzywkowski, M., (2014), Bounds on the vertex-edge domination number, C.R. Acad. Sci. Paris, Ser. I 352, pp. 363-366.
6. [6] Krishnakumari B., Venkatakrishnan Y. B. and Krzywkowski, M., (2016), On trees with total domination number equal to edge-vertex domination number plus one, Proc. Indian Acad. Sci. (Math. Sci.), 126(2), pp. 153-157.
7. [7] Lewis, J. R., Hedetniemi, S. T., Haynes, T. W. and Fricke, G. H., (2010), Vertex-edge domination, Util. Math., 81, pp. 193213.
8. [8] Peters, J. W., (1986), Theoretical and algorithmic results on domination and connectivity, Ph.D. thesis, Clemson University.