Abstract
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 $k$.
Highlights
A fundamental result of Whitney from 1933 [36] asserts that every 2-connected graph is completely characterized, up to a series of Whitney switches, by its edge set and cycles
For every vertex w ∈ V (G2), if w was adjacent to u in G, in graph G edge uw is replaced by vw
In this paper we study the algorithmic complexity of the following problem about Whitney switches
Summary
A fundamental result of Whitney from 1933 [36] asserts that every 2-connected graph is completely characterized, up to a series of Whitney switches ( known as 2-switches), by its edge set and cycles. To see the connection between Whitney switches and circular reversals of permutations, consider a cycle G with the vertices v1, . For a signed linear permutation →−π , there is an optimal sorting sequence such that no reversal cuts a signed strip.
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