Algorithm and hardness results on neighborhood total domination in graphs

TL;DR

This paper studies the computational complexity of finding minimum neighborhood total dominating sets in various graph classes, providing NP-completeness results, approximation bounds, and a linear-time algorithm for proper interval graphs.

Contribution

It extends NP-completeness to new graph classes, offers approximation bounds, and presents a linear-time algorithm for proper interval graphs.

Findings

NP-completeness for undirected path, chordal bipartite, and planar graphs

Linear-time algorithm for proper interval graphs

Approximation within O(log Δ) factor

Abstract

A set $D\subseteq V$ 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 $G$, \textsc{Min-NTDS} is the problem of finding a neighborhood total dominating set of $G$ of minimum cardinality. The decision version of \textsc{Min-NTDS} is known to be \textsf{NP}-complete for bipartite graphs and chordal graphs. In this paper, we extend this \textsf{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 $G=(V,E)$, \textsc{Min-NTDS} cannot be approximated within a factor of $(1-\varepsilon)\log |V|$, unless \textsf{NP$\subseteq$DTIME($|V|^{O(\log \log |V|)}$)} and can be approximated within a factor of $O(\log \Delta)$, where $\Delta$ is the maximum degree of the graph $G$. Finally, we show that \textsc{Min-NTDS} is \textsf{APX}-complete for graphs of degree at most $3$.