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