(Almost-)Optimal FPT Algorithm and Kernel for T-Cycle on Planar Graphs

Harmender Gahlawat, Abhishek Rathod, Meirav Zehavi

Abstract: Research of cycles through specific vertices is a central topic in graph theory. In this context, we focus on a well-studied computational problem, T-Cycle: given an undirected n-vertex graph G and a set of k vertices T ⊆ V(G) termed terminals, the objective is to determine whether G contains a simple cycle C through all the terminals. Our contribution is twofold: (i) We provide a 2^{O(√klog k)}⋅ n-time fixed-parameter deterministic algorithm for T-Cycle on planar graphs; (ii) We provide a k^{O(1)}⋅ n-time deterministic kernelization algorithm for T-Cycle on planar graphs where the produced instance is of size klog^{O(1)}k. Both of our algorithms are optimal in terms of both k and n up to (poly)logarithmic factors in k under the ETH. In fact, our algorithms are the first subexponential-time fixed-parameter algorithm for T-Cycle on planar graphs, as well as the first polynomial kernel for T-Cycle on planar graphs. This substantially improves upon/expands the known literature on the parameterized complexity of the problem.