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

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.