Turing kernelization for finding long paths in graph classes excluding a topological minor

\u3cp\u3eThe 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 boundedsize 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 k\u3csup\u3eO(1)\u3c/sup\u3e? 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\u3csub\u3e3,t\u3c/sub\u3e-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.\u3c/p\u3