A Practical Linear Time Algorithm for Optimal Tree Decomposition of Halin Graphs

TL;DR

This paper introduces extsc{H-Td}, a practical linear-time algorithm specifically designed for computing optimal tree decompositions of Halin graphs, leveraging their unique structural properties.

Contribution

The paper presents extsc{H-Td}, the first practical linear-time algorithm for optimal tree decomposition of Halin graphs, outperforming existing methods based on reduction rules.

Findings

extsc{H-Td} outperforms existing algorithms on benchmark tests.

The algorithm efficiently computes optimal decompositions for large Halin graphs.

Experimental results validate the practical effectiveness of extsc{H-Td}.

Abstract

This work proposes \textsc{H-Td}, a practical linear-time algorithm for computing an optimal-width tree decomposition of Halin graphs. Unlike state-of-the-art methods based on reduction rules or separators, \textsc{H-Td} exploits the structural properties of Halin graphs. Although two theoretical linear-time algorithms exist that can be applied to graphs of treewidth three, no practical implementation has been made publicly available. Furthermore, extending reduction-based approaches to partial $k$-trees with $k > 3$ results in increasingly complex rules that are challenging to implement. This motivates the exploration of alternative strategies that leverage structural insights specific to certain graph classes. Experimental validation against the winners of the Parameterized Algorithms and Computational Experiments Challenge (PACE) 2017 and the treewidth library \texttt{libtw} demonstrates the advantage of \textsc{H-Td} when the input is known to be a Halin graph.