A Linear Kernel for Planar Total Dominating Set

TL;DR

This paper presents an explicit linear kernel with at most 410k vertices for the Total Dominating Set problem on planar graphs, improving understanding of kernelization for domination problems in planar graph classes.

Contribution

It provides a concrete, efficiently constructible linear kernel for Total Dominating Set on planar graphs with explicit reduction rules and constants.

Findings

Kernel size is at most 410k vertices.

Constructive reduction rules are explicitly described.

Extends kernelization techniques to Total Dominating Set on planar graphs.

Abstract

A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set.