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.
@article{10_37236_11119,
author = {Cheolwon Heo and Hyobin Kim and Siggers Mark},
title = {The complexity of the matroid homomorphism problem},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {2},
doi = {10.37236/11119},
zbl = {1516.05021},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11119/}
}
TY - JOUR
AU - Cheolwon Heo
AU - Hyobin Kim
AU - Siggers Mark
TI - The complexity of the matroid homomorphism problem
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/11119/
DO - 10.37236/11119
ID - 10_37236_11119
ER -
%0 Journal Article
%A Cheolwon Heo
%A Hyobin Kim
%A Siggers Mark
%T The complexity of the matroid homomorphism problem
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/11119/
%R 10.37236/11119
%F 10_37236_11119
Cheolwon Heo; Hyobin Kim; Siggers Mark. The complexity of the matroid homomorphism problem. The electronic journal of combinatorics, Tome 30 (2023) no. 2. doi: 10.37236/11119
