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Bounds on Neighborhood Total Domination Numberin Graphs

  • Kan WangEmail author
  • Changhong Lu
  • Bing Wang
Original Paper
  • 21 Downloads

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.

Keywords

Neighborhood total domination Upper bounds Size Order 

Mathematics Subject Classification

05C15 

Notes

Acknowledgements

The authors would like to thank the referees for many constructive suggestions on the revision of this paper.

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Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Xingzhi CollegeZhejiang Normal UniversityJinhuaChina
  2. 2.Department of Mathematics, Shanghai key laboratory of PMMPEast China Normal UniversityShanghaiChina

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