Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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THE OUTER-CONNECTED VERTEX EDGE DOMINATION NUMBER OF A TREE

  • Received : 2015.12.17
  • Accepted : 2017.11.30
  • Published : 2018.01.31
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Abstract

For a given graph G = (V, E), a set $D{\subseteq}V(G)$ is said to be an outer-connected vertex edge dominating set if D is a vertex edge dominating set and the graph $G{\backslash}D$ is connected. The outer-connected vertex edge domination number of a graph G, denoted by ${\gamma}^{oc}_{ve}(G)$, is the cardinality of a minimum outer connected vertex edge dominating set of G. We characterize trees T of order n with l leaves, s support vertices, for which ${\gamma}^{oc}_{ve}(T)=(n-l+s+1)/3$ and also characterize trees with equal domination number and outer-connected vertex edge domination number.

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References

  1. R. Boutrig, M. Chellalli, T. Haynes, and S. T. Hedetniemi, Vertex-edge domination in graphs, Aequationes Math. 90 (2016), no. 2, 355-366. https://doi.org/10.1007/s00010-015-0354-2
  2. J. Cyman, The outer-connected domination number of a graph, Australas. J. Combin. 38 (2007), 35-46.
  3. T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Fundamentals of Domination in Graphs, Monographs and Textbooks in Pure and Applied Mathematics, 208, Marcel Dekker, Inc., New York, 1998.
  4. B. Krishnakumari, Y. B. Venkatakrishnan, and M. Krzywkowski, Bounds on the vertex-edge domination number of a tree, C. R. Math. Acad. Sci. Paris 352 (2014), no. 5, 363-366. https://doi.org/10.1016/j.crma.2014.03.017
  5. J. Lewis, S. Hedetniemi, T. Haynes, and G. Fricke, Vertex-edge domination, Util. Math. 81 (2010), 193-213.
  6. J. Peters, Theoretical and Algorithmic Results on Domination and Connectivity, Ph.D. Thesis, Clemson University, 1986.