Abstract
A dominating set D of G is a subset of V(G), such that every vertex in \(V(G){\setminus } D\) is adjacent to at least one vertex in D. A neighborhood total dominating set, abbreviated for NTD set D, is a dominating set of G with an extra property: the subgraph induced by the open neighborhood of D, denoted by G[N(D)], has no isolated vertices. The neighborhood total domination number, denoted by \(\gamma _{nt}(G)\), is the minimum cardinality of an NTD set in G. A classical result of Vizing relates the size and the domination number of a graph of given order. In this paper, we present a Vizing-like result for \(\gamma _{nt}(G)\). Some results for \(\gamma _{nt}(G)\) in terms of other graphic parameters, such as girth, diameter, and degree of G, are also obtained.
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References
Arumugam, S., Sivagnanam, C.: Neighborhood total domination in graphs. Opusc. Math. 31, 519–531 (2011)
Cockayne, E.J., Dawes, R.M., Hedetniemi, S.T.: Total domination in graph. Networks 10, 211–219 (1980)
Dankelmann, P., Domke, G.S., Goddard, W., Grobler, P., Hattingh, J.H., Swart, H.C.: Maximum sizes of graphs with given domination parameters. Discrete Math. 281, 137–148 (2004)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker Inc, New York (1998)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker Inc, New York (1998)
Henning, M.A., Yeo, A.: Total domination in graphs. Springer, New York (2013)
Henning, M.A., Rad, N.J.: Bounds on neighborhood total domination in graphs. Discrete Appl. Math. 161, 2460–2466 (2013)
Lu, C.H., Wang, B., Wang, K.: Algorithm complexity of neighborhood total domination and \((\rho, \gamma _{nt})\)-graphs. J. Comb. Optim. 35, 1–12 (2018)
Sanchis, L.A.: Maximum number of edges in connected graphs with a given domination number. Discrete Math. 87, 65–72 (1991)
Vizing, V.G.: A bound on the external satbility number of a graph. Dokl. Akad. Nauk SSSR. 164, 729–731 (1965)
West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)
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The authors would like to thank the referees for many constructive suggestions on the revision of this paper.
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Communicated by Hossein Hajiabolhassan.
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Supported in part by National Natural Science Foundation of China (Nos. 11371008 and 11871222) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).
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Wang, K., Lu, C. & Wang, B. Bounds on Neighborhood Total Domination Numberin Graphs. Bull. Iran. Math. Soc. 45, 1135–1143 (2019). https://doi.org/10.1007/s41980-018-0189-4
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DOI: https://doi.org/10.1007/s41980-018-0189-4