The complexity of matroid homomorphism reconfiguration
Cheolwon Heo, Mark Siggers
TL;DR
This work studies the reconfiguration of matroid homomorphisms ${\rm Recol}_\mathbb{M}(N)$ for binary matroids, defining adjacency via cocircuits and establishing a Recolouring framework analogous to graphs. The authors develop decision graphs $D(N)$ and their Tutte-inspired connections to bridge matroid recolouring with graph recolouring, and they introduce Kempe recolouring to transfer complexity results between settings. They prove a dichotomy: ${\rm Recol}_\mathbb{M}(N)$ is trivial when $N$ dismantles to $M(K_2)$ (or to a loop), and PSPACE-complete for $N=M(K_n)$ with $n\ge 3$, and for any graphic matroid containing $M(K_5)$, via gadget constructions and reductions. The work furthermore frames a conjecture that triviality corresponds precisely to dismantling to $M(K_2)$ and raises open questions about non-graphic matroids and girth-based tractability. This connects matroid theory, Tutte duality, and reconfiguration complexity, broadening understanding of colourings beyond graphs. $${\rm Recol}_\mathbb{M}(N)$$ thus inherits rich structure from graphic cases and Tutte-transformations, with clear avenues for future exploration.
Abstract
We consider a reconfiguration version of the homomorphism problem ${\rm Hom}_\mathbb{M}(N)$ for binary matroids $N$. This reconfiguration problem, ${\rm Recol}_\mathbb{M}(N)$, asks, for two homomorphisms $φ$ and $ψ$ of a matroid $M$ to $N$, if there is a path of homomorphism from $φ$ to $ψ$ such that consecutive homomorphism in the path differ on a single cocircuit of $N$. We show that this problem is trivial in the case that $N$ dismantles to the graphic matroid $M(K_2)$, and that the problem is ${\rm PSPACE}$-complete when $N$ is the graphic matroid $M(K_3)$, $M(K_4)$, or any graphic matroid containing $M(K_5)$.