Journal of Combinatorial Optimization

, Volume 35, Issue 2, pp 424–435 | Cite as

Algorithm complexity of neighborhood total domination and \((\rho ,\gamma _{nt})\)-graphs

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Abstract

A neighborhood total dominating set, abbreviated for NTD-set D, is a vertex set of G such that D is a dominating set with an extra property: the subgraph induced by the open neighborhood of D has no isolated vertex. The neighborhood total domination number, denoted by \(\gamma _{nt}(G)\), is the minimum cardinality of a NTD-set in G. In this paper, we prove that NTD problem is NP-complete for bipartite graphs and split graphs. Then we give a linear-time algorithm to determine \(\gamma _{nt}(T)\) for a given tree T. Finally, we characterize a constructive property of \((\gamma _{nt},2\gamma )\)-trees and provide a constructive characterization for \((\rho ,\gamma _{nt})\)-graphs, where \(\gamma \) and \(\rho \) are domination number and packing number for the given graph, respectively.

Keywords

Neighborhood total domination Algorithm complexity Labeled graph 

Mathematics Subject Classfication

05C15 

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Shanghai Key Laboratory of PMMPEast China Normal UniversityShanghaiPeople’s Republic of China

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