SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 1298-1336, January 2021.
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 \sf NP-complete for cycles, and we investigate its parameterized complexity. We show that the problem admits a kernel of size $\mathcal{O}(k)$ 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

SIAM Journal on Discrete Mathematics，第 35 卷，第 2 期，第 1298-1336 页，2021 年 1 月
。 1933 年惠特尼的基本定理断言 2-连通图 $G$ 和 $H$ 是 2-同构的，或者等效地，它们的循环拟阵是同构的，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，我们能否通过应用至多 $k$ 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 \sf NP-complete，我们研究了它的参数化复杂度。我们表明，该问题承认一个大小为 $\mathcal{O}(k)$ 的核，因此当由 $k$ 参数化时是固定参数易处理的。