Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

DOUBLE VERTEX-EDGE DOMINATION IN TREES

  • Chen, Xue-Gang (Department of Mathematics North China Electric Power University) ;
  • Sohn, Moo Young (Department of Mathematics Changwon National University)
  • Received : 2021.03.03
  • Accepted : 2021.10.14
  • Published : 2022.01.31
Download PDF
⟨ Previous Next ⟩

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 called a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S. The minimum cardinality of a double vertex-edge dominating set of G is the double vertex-edge domination number γdve(G). In this paper, we provide an upper bound on the double vertex-edge domination number of trees in terms of the order n, the number of leaves and support vertices, and we characterize the trees attaining the upper bound. Finally, we design a polynomial time algorithm for computing the value of γdve(T) for any trees. This gives an answer of an open problem posed in [4].

Keywords

Acknowledgement

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2020R1I1A3A04036669).

References

  1. R. Boutrig, M. Chellali, T. W. 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. E. J. Cockayne, S. Goodman, and S. T. Hedetniemi, A linear algorithm for the domination number of a tree, Inform. Process. Lett. 4 (1975) 41-44. https://doi.org/10.1016/0020-0190(75)90011-3
  3. S. T. Hedetniemi, R. Laskar, and J. Pfaff, A linear algorithm for finding a minimum dominating set in a cactus, Discrete Appl. Math. 13 (1986), no. 2-3, 287-292. https://doi.org/10.1016/0166-218X(86)90089-2
  4. B. Krishnakumari, M. Chellali, and Y. B. Venkatakrishnan, Double vertex-edge domination, Discrete Math. Algorithms Appl. 9 (2017), no. 4, 1750045, 11 pp. https://doi.org/10.1142/S1793830917500458
  5. 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
  6. J. Lewis, S. T. Hedetniemi, T. W. Haynes, and G. H. Fricke, Vertex-edge domination, Util. Math. 81 (2010), 193-213.
  7. J. W. Peters, Theoretical and algorithmic results on domination and connectivity, Ph.D. thesis, Clemson University, 1986.
  8. Y. Zhao, Z. Liao, and L. Miao, On the algorithmic complexity of edge total domination, Theoret. Comput. Sci. 557 (2014), 28-33. https://doi.org/10.1016/j.tcs.2014.08.005