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 nonnegative integer k, decide whether the cyclability of G is at least k, is NP-hard. We study the parametrized complexity of this problem. We prove that this problem, parameterized by k, is co-W[1]-hard and that it does not admit a polynomial kernel on planar graphs, unless NP ⊆ co-NP=poly. On the positive side, we give an FPT algorithm for planar graphs that runs in time 22O(k2 log k). n2. Our algorithm is based on a series of graph-theoretical results on cyclic linkages in planar graphs. [ABSTRACT FROM AUTHOR]