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

Author

Listed:
  • Changhong Lu

    (East China Normal University)

  • Bing Wang

    (East China Normal University)

  • Kan Wang

    (East China Normal University)

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.

Suggested Citation

  • Changhong Lu & Bing Wang & Kan Wang, 2018. "Algorithm complexity of neighborhood total domination and $$(\rho ,\gamma _{nt})$$ ( ρ , γ n t ) -graphs," Journal of Combinatorial Optimization, Springer, vol. 35(2), pages 424-435, February.
    • Handle: RePEc:spr:jcomop:v:35:y:2018:i:2:d:10.1007_s10878-017-0181-6
    DOI: 10.1007/s10878-017-0181-6
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