Contraction–deletion algorithm for non-graphic matroid intersection products

Construct a contraction–deletion algorithm for determining the tropical intersection product $(-\Trop(M))\cdot\Trop(N)\cdot\Trop(y_{\epsilon}-1)$ when M and N are matroids on a common ground set that are not both graphic matroids.

Background

The paper extends the tropical formulation of planar realisation numbers to a bigraph setting, where two graphs share a common edge set. In this setting, the relevant quantity is expressed as a tropical intersection product involving the flipped Bergman fan of one graphic matroid, the Bergman fan of the other, and a tropical hyperplane.

Combining this formulation with an existing realisation-number algorithm yields an inductive contraction–deletion procedure when both matroids are graphic. The unresolved question is whether an analogous algorithm can be developed for arbitrary non-graphic matroids on a common ground set.