Algorithm complexity of neighborhood total domination and $$(\rho,\gamma _{nt})$$ ( ρ, γ n t ) -graphs

Changhong Lu (), Bing Wang () and Kan Wang ()
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Changhong Lu: East China Normal University
Bing Wang: East China Normal University
Kan Wang: East China Normal University

Journal of Combinatorial Optimization, 2018, vol. 35, issue 2, No 8, 424-435

Abstract: 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 $$\gamma _{nt}(G)$$ γ n t ( 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 $$\gamma _{nt}(T)$$ γ n t ( T ) for a given tree T. Finally, we characterize a constructive property of $$(\gamma _{nt},2\gamma )$$ ( γ n t , 2 γ ) -trees and provide a constructive characterization for $$(\rho ,\gamma _{nt})$$ ( ρ , γ n t ) -graphs, where $$\gamma $$ γ and $$\rho $$ ρ are domination number and packing number for the given graph, respectively.

Keywords: Neighborhood total domination; Algorithm complexity; Labeled graph; 05C15 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10878-017-0181-6

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