Algorithm and hardness results on neighborhood total domination in graphs
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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(loglog |V|)) and can be approximated within a factor of O(logΔ), 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.
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