Algorithm and hardness results on neighborhood total domination in graphs

Abstract

A set D⊆V of a graph G=(V,E) is called a neighborhood total dominating set of G if D is a dominating set and the subgraph of G induced by the open neighborhood of D has no isolated vertex. Given a graph G, Min-NTDS is the problem of finding a neighborhood total dominating set of G of minimum cardinality. The decision version of Min-NTDS is known to be NP-complete for bipartite graphs and chordal graphs. In this paper, we extend this NP-completeness result to undirected path graphs, chordal bipartite graphs, and planar graphs. We also present a linear time algorithm for computing a minimum neighborhood total dominating set in proper interval graphs. We show that for a given graph G=(V,E), Min-NTDS cannot be approximated within a factor of (1−ε)log⁡|V|, unless NP⊆DTIME(|V|O(log⁡log⁡|V|)) and can be approximated within a factor of O(log⁡Δ), where Δ is the maximum degree of the graph G. Finally, we show that Min-NTDS is APX-complete for graphs of degree at most 3.