Kernelization of Whitney Switches

TL;DR

This paper investigates the parameterized complexity of transforming 2-isomorphic graphs via Whitney switches, proving the problem is fixed-parameter tractable with a kernel of size proportional to the number of switches allowed.

Contribution

It introduces a kernelization result for the Whitney switch transformation problem, establishing fixed-parameter tractability based on the number of switches.

Findings

The problem admits a kernel of size O(k).

The problem is fixed-parameter tractable when parameterized by k.

The problem is NP-complete for cycles.

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 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.