Kernelization of Whitney Switches

A fundamental theorem of Whitney from 1933 asserts that 2-connected graphs $G$ and $H$ are 2-isomorphic, or equivalently, their cycle matroids isomorphic if only can be transformed into by a series operations called switches. In this paper we consider the quantitative question arising Whitney's theorem: Given two 2-isomorphic graphs, transform one another applying at most $k$ switches? This problem is already \sf NP-complete for cycles, investigate its parameterized complexity. We show admits kernel size $\mathcal{O}(k)$ thus fixed-parameter tractable when $k$.