The Complexity of the Matroid Homomorphism Problem

• Published in: The Electronic journal of combinatorics Vol. 30; no. 2
• Main Authors: Heo, Cheolwon, Kim, Hyobin, Mark, Siggers
• Format: Journal Article
• Language: English
• Published: 19.05.2023
• ISSN: 1077-8926; 1077-8926
• DOI: 10.37236/11119

We show that for every binary matroid $N$ there is a graph $H_*$ such that for the graphic matroid $M_G$ of a graph $G$, there is a matroid-homomorphism from $M_G$ to $N$ if and only if there is a graph-homomorphism from $G$ to $H_*$. With this we prove a complexity dichotomy for the problem $\rm{Hom}_\mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $\rm{NP}$-complete. We also get dichotomies for the list, extension, and retraction versions of the problem.