# Is there a version of Robertson-Seymour's graph minor theorem for odd minors?

A graph $H$ is an odd minor of a graph $G$ if $H$ arises from $G$ by first deleting some vertices and edges and then contracting all edges in some edge cut.

Is it known that families of graphs that are characterized by a list of forbidden odd minors are characterized by a finite list of forbidden odd minors?

• I am pretty sure that the well-quasi-ordering result of binary matroids would imply the well-quasi-ordering of graphs under odd minors. I also remember hearing Tony Huynh talking on structure of "group-labelled" graphs with a forbidden minors. Perhaps, though not everything is published, it must be known. – Sang-il Aug 9 '17 at 2:56

Also, for any fixed signed graph $H$, there is a polynomial-time algorithm to test if an input signed graph contains $H$ as a minor. Together with the positive answer to your question, this implies that there is a polynomial-time algorithm to test for any minor-closed property of signed graphs. See my PhD thesis for the algorithm (which works more generally for any $\Gamma$-labelled graph, where $\Gamma$ is a fixed finite abelian group).
• If $H$ is an odd minor of $G$, then $(H, E(H))$ is a signed-minor of $(G, E(G))$. – Tony Huynh Aug 14 '17 at 18:08