The Parameterized Complexity of Graph Cyclability

The cyclability of a graph is the maximum integer k for which every k vertices lie on a cycle. The algorithmic version of the problem, given a graph G and a non-negative integer k, decide whether the cyclability of G is at least k, is NP-hard. We prove that this problem, parameterized by k, is co−W[1]-hard. We give an FPT algorithm for planar graphs that runs in time \(2^{2^{O(k^2\log k)}}\cdot n^2\). Our algorithm is based on a series of graph theoretical results on cyclic linkages in planar graphs.

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