Algorithm complexity of neighborhood total domination and (ρ,γnt)-graphs - Lu, Changhong; Wang, Bing; Wang, Kan (Journal of combinatorial optimization, 2018)
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 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 γ n t ( T ) for a given tree T . Finally, we characterize a constructive property of ( γ n t , 2 γ ) -trees and provide a constructive characterization for ( ρ , γ n t ) -graphs, where γ and ρ are domination number and packing number for the given graph, respectively.