Abstract
Let p be a positive integer and G = (V, E) be a simple graph. A p-dominating set of G is a subset \({D\,{\subseteq}\, V}\) such that every vertex not in D has at least p neighbors in D. The p-domination number of G is the minimum cardinality of a p-dominating set of G. The p-bondage number of a graph G with (ΔG) ≥ p is the minimum cardinality among all sets of edges \({B\subseteq E}\) for which γ p (G − B) > γ p (G). For any integer p ≥ 2 and tree T with (ΔT) ≥ p, this paper shows that 1 ≤ b p (T) ≤ (ΔT) − p + 1, and characterizes all trees achieving the equalities.
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References
Blidia M., Chellali M., Favaron O.: Independence and 2-domination in trees. Aust. J. Combin. 33, 317–327 (2005)
Blidia M., Chellali M., Volkmann L.: Some bounds on the p-domination number in trees. Discrete Math. 306, 2031–2037 (2006)
Carlson K., Develin M.: On the bondage number of planar and directed graphs. Discrete Math. 306(8–9), 820–826 (2006)
Caro Y., Roditty Y.: A note on the k-domination number of a graph. Int. J. Math. Sci. 13, 205–206 (1990)
Chartrant G., Lesniak L.: Graphs & Digraphs, 3rd edn. Chapman & Hall, London (1996)
Dunbar, J.E., Haynes, T.W., Teschner, U., Volkmann, L.: Bondage, insensitivity, and reinforcement. Domination in Graphs: Advanced Topics (Haynes, T.W., Hedetniemi, S.T., Slater, P.J. eds.), 471–489, Monogr. Textbooks Pure Appl. Math., 209, Marcel Dekker, New York, (1998)
Favaron O.: On a conjecture of Fink and Jacobson concerning k-domination and k-dependence. J. Combin. Theory Ser. B 39, 101–102 (1985)
Fink J.F., Jacobson M.S.: n-Domination in graphs. In: Alavi, Y., Schwenk, A.J. (eds) Graph Theory with Applications to Algorithms and Computer Science, pp. 283–300. Wiley, New York (1985)
Fink J.F., Jacobson M.S., Kinch L.F., Roberts J.: The bondage number of a graph. Discrete Math. 86, 47–57 (1990)
Fischermann M., Rautenbach D., Volkmann L.: Remarks on the bondage number of planar graphs. Discrete Math. 260, 57–67 (2003)
Haynes T.W., Hedetniemi S.T., Slater P.J.: Fundamentals of Domination in Graphs. Marcel Deliker, New York (1998)
Haynes T.W., Hedetniemi S.T., Slater P.J.: Domination in Graphs: Advanced Topics. Marcel Deliker, New York (1998)
Hartnell B.L., Rall D.F.: A characterization of trees in which no edge is essential to the domination number. Ars Combin. 33, 65–76 (1992)
Hartnell B.L., Rall D.F.: Bounds on the bondage number of a graph. Discrete Math. 128, 173–177 (1994)
Hattingh J.H., Plummer A.R.: Restrained bondage in graphs. Discrete Math. 308(23), 5446–5453 (2008)
Huang J., Xu J.-M.: The bondage numbers of extended de Bruijn and Kautz digraphs. Comput Math Appl 51(6–7), 1137–1147 (2006)
Huang J., Xu J.-M.: The total domination and bondage numbers of extended de bruijn and Kautz digraphs. Comput Math Appl 53(8), 1206–1213 (2007)
Huang J., Xu J.-M.: The bondage number of graphs with small crossing number. Discrete Math. 307(14), 1881–1897 (2007)
Huang J., Xu J.-M.: The bondage numbers and efficient dominations of vertex-transitive graphs. Discrete Math. 308(4), 571–582 (2008)
Kang L.-Y., Sohn M.Y., Kim H.K.: Bondage number of the discrete torus C n × C 4. Discrete Math. 303, 80–86 (2005)
Kang L., Yuan J.: Bondage number of planar graphs. Discrete Math. 222, 191–198 (2000)
Liu H., Sun L.: The bondage and connectivity of a graph. Discrete Math. 263, 289–293 (2003)
Raczek J.: Paired bondage in trees. Discrete Math. 308(23), 5570–5575 (2008)
Teschner U.: New results about the bondage number of a graph. Discrete Math. 171, 249–259 (1997)
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The work was supported by NNSF of China (No. 10671191, 10701068).
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Lu, Y., Xu, JM. The p-Bondage Number of Trees. Graphs and Combinatorics 27, 129–141 (2011). https://doi.org/10.1007/s00373-010-0956-3
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DOI: https://doi.org/10.1007/s00373-010-0956-3