Linear-Time FPT Algorithm for Surface Disjoint Paths via Surface Cutting

2026-09-24

AI Summary

This study addresses the problem of finding $k$ vertex-disjoint paths in graphs embedded on surfaces of bounded Euler genus. Methodologically, it extends recent advances for planar graphs to surface-embedded graphs by proposing an efficient graph decomposition strategy based on surface cutting. This approach integrates boundary complexity control with planar linkage compression techniques to handle path connectivity in such embeddings. The primary contributions include a linear-time fixed-parameter tractable (FPT) algorithm with a running time of $2^{O(k^2+g^2)}n$, as well as the derivation of two polynomial-size kernels. Collectively, these results establish an efficient parameterized framework for solving disjoint paths problems on surface-embedded graphs.

Abstract

We study the \textsc{$k$-Disjoint Paths} problem on a graph embedded on a surface with bounded Euler genus. Given a graph $G$ with $n$ vertices and $k$ vertex pairs embedded on a surface of Euler genus $g$, we present a $2^{O(k^2+g^2)}n$-time algorithm that computes $k$ pairwise vertex-disjoint paths connecting the given vertex pairs if such paths exist. Our approach relies on the decomposition of $G$ into $O(k+g)$ planar subgraphs while bounding the complexity of the boundaries between these subgraphs. This approach enables the use of techniques for compressing linkages in planar graphs. Moreover, our techniques yield two kernels of size polynomial in $k$, $g$, and the treewidth of the graph, and of size $2^{O(k+g)}$. These results extend recent advances on \textsc{$k$-Disjoint Paths} on planar graphs [Cho et al. SODA 2023] and [Włodarczyk and Zehavi FOCS 2023] to surface-embedded graphs.

Problem

• k-Disjoint Paths
• Surface-embedded graphs
• Euler genus
• Parameterized complexity

Innovation

• Fixed-Parameter Tractable
• Disjoint Paths
• Surface-Embedded Graphs
• Kernelization
• Linkage Compression