Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor - Jansen, Bart M. P.; Pilipczuk, Marcin; Wrochna, Marcin (Algorithmica, 2019)
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 k O ( 1 ) ? 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.