Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor

TL;DR

This paper proves that the NP-hard problem of finding long paths in certain graph classes can be efficiently solved using Turing kernelization, especially in graphs excluding a fixed topological minor or with a bounded modulator.

Contribution

It establishes the existence of polynomial Turing kernels for k-Path in H-topological-minor-free graphs and graphs with a polynomially bounded vertex modulator, extending previous results.

Findings

Polynomial Turing kernel for k-Path in H-topological-minor-free graphs.

Polynomial Turing kernel for graphs with a known vertex modulator.

Utilizes graph minors decomposition to enable safe reductions.

Abstract

The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k-Path admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size poly(k)? We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and $K_{3,t}$-minor-free graphs. Moreover, we show that k-Path even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minor-free graph that does not contain a k-path, has a separation that can safely be reduced after communication with the oracle.