By signed graph, I mean each edge is designated either odd or even (e.g. as in Guenin's result for weakly bipartite graphs).



Does anyone know a specific polynomial-time algorithm to detect if a given signed graph contains an odd-K4 as a signed minor?
Yes, in my [PhD thesis](https://uwspace.uwaterloo.ca/handle/10012/4716), we prove the stronger result that for any fixed signed graph $(G, \Sigma)$, there is a polynomial-time algorithm to test if an input signed graph contains a $(G, \Sigma)$-minor. Actually, we prove the stronger result that for any abelian group $\Gamma$, testing for a fixed $\Gamma$-labelled minor can be done in polynomial time (signed graphs correspond to the case $\Gamma=\mathbb{Z} / 2 \mathbb{Z}$). 


As a word of warning, the algorithm is rather complicated. For example, the special case that $\Gamma$ is the trivial group includes minor testing (for graphs), which requires much of the Graph Minors machinery developed by Robertson and Seymour. 


If you only care about odd-$K\_4$-minors, there is an easier algorithm of Kawarabayashi, Li and Reed [here](http://epubs.siam.org/doi/abs/10.1137/1.9781611973075.27).


