Algorithm complexity of neighborhood total domination and (ρ, γ_nt)-graphs

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 γ_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 γ_nt (T) for a given tree T. Finally, we characterize a constructive property of (γ_nt, 2γ)-trees and provide a constructive characterization for (ρ, γ_nt)-graphs, where γ and ρ are domination number and packing number for the given graph, respectively.