Total domination and vertex-edge domination in trees.
AbstractA 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.
R. Boutrig, M. Chellali, T. W. Haynes and S. T. Hedetniemi, Vertexedge domination in graphs. Aequat. Math., 90, pp. 355—366, (2016).
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).
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).
J. R. Lewis, S. T. Hedetniemi, T. W. Haynes and G. H. Fricke, Vertexedge domination. Util. Math. 81, pp. 193—213, (2010).
J. W. Peters. Theoretical and algorithmic results on domination and connectivity. Ph.D. Thesis, Clemson University, (1986).
Copyright (c) 2019 Proyecciones. Journal of Mathematics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.