Semitotal domination in unit disk graphs

Abstract

A set $S\subseteq V$ is called a {\em semitotal dominating set} of $G=(V,E)$ if every vertex in $V\setminus S$ is adjacent to at least one vertex in $S$, and every vertex in $S$ is within distance 2 of another vertex in $S$. The corresponding decision problem is NP-complete even for unit disk graphs. In this paper, we present a 5-factor approximation algorithm for the Minimum Semitotal Domination problem on unit disk graphs in the graph-based input model. The algorithm processes the layers of a Breadth-First-Search tree and constructs a maximal independent set whose vertices satisfy the semitotal condition. For a graph with $n$ vertices and $m$ edges, the algorithm runs in $O(n+m)$ time, and hence in $O(n^2)$ time in the worst case. This improves the previously known 5.75-approximation algorithm with $O(n^3)$ running time.

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