A 2^O(k)n algorithm for k-cycle in minor-closed graph families

Let C be a proper minor-closed family of graphs. We present a randomized algorithm that given a graph G∈C with n vertices, finds a simple cycle of size k in G (if exists) in 2^O(k)n time. The algorithm applies to both directed and undirected graphs. In previous linear time algorithms for this problem, the runtime dependence on k is super-exponential. The algorithm can be derandomized yielding a 2^O(k)nlog⁡n time algorithm.