Abstract
A vertex subset S of a graph G is a 2-dominating set of G if every vertex not in S is adjacent to two vertices of S. The 2-domination number \(\gamma _2(G)\) is the minimum cardinality of a 2-dominating set of G. The 2-reinforcement number \(r_2(G)\) is the smallest number of extra edges whose addition to G results in a graph \(G'\) with \(\gamma _2(G')< \gamma _2(G)\). Let T be a tree. It is showed by Lu, Hu, and Xu that \(r_2(T)\le 3\). In this paper, we will show that \(r_2(T)=3\) if and only if there is a 2-dominating set S of T such that T contains neither S-vulnerable vertices nor S-vulnerable paths.
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References
Aram, H., Sheikholeslami, S.M., Volkmann, L.: On the total \(\{k\}\)-domination and total \(\{k\}\)-domatic number of graphs. Bull. Malays. Math. Sci. Soc. (2) 36(1), 39–47 (2013)
Aram, H., Sheikholeslami, S.M., Volkmann, L.: Signed \(k\)-domatic numbers of digraphs. Bull. Malays. Math. Sci. Soc. (2) 36(1), 143–150 (2013)
Blidia, M., Chellali, M., Favaron, O.: Independence and 2-domination in trees. Australas. J. Comb. 33, 317–327 (2005)
Blidia, M., Chellali, M., Volkmann, L.: Some bounds on the \(p\)-domination number in trees. Discret. Math. 306, 2031–2037 (2006)
Blair, J.R.S., Goddard, W., Hedetniemi, S.T., Horton, S., Jones, P., Kubicki, G.: On domination and reinforcement numbers in trees. Discret. Math. 308, 1165–1175 (2008)
Bondy, J.A., Murty, U.S.R.: Graph theory. In: Axler, S., Ribert, K.A. (eds.) GTM, vol. 244, Springer, New York (2008)
Chellali, M., Favaron, O., Hansberg, A., Volkmann, L.: \(k\)-Domination and \(k\)-independence in graphs: a survey. Graphs Comb. 28(1), 1–55 (2012)
DeLaViña, E., Goddard, W., Henning, M.A., Pepper, R., Vaughan, E.R.: Bounds on the \(k\)-domination number of a graph. Appl. Math. Lett. 24, 996–998 (2011)
Favaron, O.: On a conjecture of Fink and Jacobson concerning \(k\)-dependence. J. Comb. Theory Ser. B 39, 101–102 (1985)
Favaron, O., Hansberg, A., Volkmann, L.: On \(k\)-domination and minimum degree in graphs. J. Graph Theory 57, 33–40 (2008)
Fink, J.F., Jacobson, M.S.: \(n\)-Domination in graphs. In: Alavi, Y., Chartrand, G., Lesniak, L., Lick, D.R., Wall, C.E. (eds.) Graph Theory with Applications to Algorithms and Computer Science, pp. 283–300. Wiley, New York (1985)
Hansberg, A., Meierling, D., Volkmann, L.: Independence and \(k\)-domination in graphs. Int. J. Comput. Math. 88(5), 905–915 (2011)
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)
Henning, M.A., Rad, N.J., Raczek, J.: A note on total reinforcement in graph. Discret. Appl. Math. 159, 1443–1446 (2011)
Hu, F.-T., Xu, J.-M.: On the complexity of the bondage and reinforcement problems. J. Complex. 28(2), 192–201 (2011)
Huang, J., Wang, J.W., Xu, J.-M.: Reinforcement number of digraphs. Discret. Appl. Math. 157, 1938–1946 (2009)
Karami, H., Khoeilar, R., Sheikholeslami, S.M., Khodkar, A.: Global signed domination in graphs. Bull. Malays. Math. Sci. Soc. (2) 36(2), 363–372 (2013)
Kok, J., Mynhardt, C.M.: Reinforcement in graphs. Congr. Numer. 79, 225–231 (1990)
Lu, Y., Hu, F.-T., Xu, J.-M.: On the \(p\)-reinforcement and the complexity. J. Comb. Optim. (2013). doi:10.1007/s10878-013-9597-9
Lu, Y., Hou, X.-M., Xu, J.-M., Li, N.: Trees with unique minimum \(p\)-dominating sets. Util. Math. 86, 193–205 (2011)
Li, H.Z., Li, X.L., Mao, Y.P.: On extremal graphs with at most two internally disjoint Steiner trees connecting any three vertices. Bull. Malays. Math. Sci. Soc. (2) 37(3), 747–756 (2014)
Lu, Y., Xu, J.-M.: Trees with maximum \(p\)-reinforcement number. http://arxiv.org/abs/1211.5742v1
Wu, Y.J., Yu, Q.L.: A characterization of graphs with equal domination number and vertex cover number. Bull. Malays. Math. Sci. Soc. (2) 35(3), 803–806 (2012)
Zhang, J.H., Liu, H.L., Sun, L.: Independence bondage and reinforcement number of some graphs. Trans. Beijin Inst. Technol. 23, 140–142 (2003)
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The work was supported by NNSF of China (No. 11201374) and the Fundamental Research Funds for the Central University (NO. 3102014JCQ01074).
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Communicated by Xueliang Li.
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Lu, Y., Song, W. & Yang, HL. Trees with 2-Reinforcement Number Three. Bull. Malays. Math. Sci. Soc. 39, 821–838 (2016). https://doi.org/10.1007/s40840-015-0140-2
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DOI: https://doi.org/10.1007/s40840-015-0140-2