Bounds on global secure sets in cactus trees

Katarzyna Jesse-Józefczyk 1
  • 1 University of Zielona Góra


Let G = (V, E) be a graph. A global secure set SD ⊆ V is a dominating set which satisfies the condition: for all X ⊆ SD, |N[X] ∩ SD| ≥ | N[X] − SD|. A global defensive alliance is a set of vertices A that is dominating and satisfies a weakened condition: for all x ∈ A, |N[x] ∩ A| ≥ |N[x] − A|. We give an upper bound on the cardinality of minimum global secure sets in cactus trees. We also present some results for trees, and we relate them to the known bounds on the minimum cardinality of global defensive alliances.

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  • [1] Brigham R.C., Dutton R.D., Hedetniemi S.T., Security in graphs, Discrete Appl. Math., 2007, 155(13), 1708–1714

  • [2] Cami A., Balakrishnan H., Deo N., Dutton R.D., On the complexity of finding optimal global alliances, J. Combin. Math. Combin. Comput., 2006, 58, 23–31

  • [3] Chellali M., Haynes T.W., Global alliances and independence in trees, Discuss. Math. Graph Theory, 2007, 27(1), 27, 19–27

  • [4] Eroh L., Gera R., Global alliance partition in trees, J. Combin. Math. Combin. Comput., 2008, 66, 161–169

  • [5] Haynes T.W., Hedetniemi S.T., Henning M.A., Global defensive alliances in graphs, Electron. J. Combin., 2003, 10(1), #R47

  • [6] Kristiansen P., Hedetniemi S.M., Hedetniemi S.T., Alliances in graphs, J. Combin. Math. Combin. Comput., 2004, 48, 157–177


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