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

中文翻译：

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惠特尼开关的内核化

1933 年惠特尼的一个基本定理断言 2-连通图 G 和 H 是 2-同构的，或者等价地，它们的圈拟阵是同构的，当且仅当 G 可以通过一系列称为惠特尼开关的操作转换为 H。在本文中，我们考虑惠特尼定理提出的定量问题：给定两个 2-同构图，我们能否通过应用最多 k 个惠特尼开关将一个图转换为另一个图？这个问题对于循环来说已经是 NP 完全的，我们研究了它的参数化复杂度。我们表明该问题允许一个大小为 O(k) 的核，因此，当由 k 参数化时，它是固定参数易处理的。