Edge cut splitting formulas for Tutte-Grothendieck invariants

Tutte-Grothendieck invariants of graphs are mappings from a class of graphs to a commutative ring that are characterized recursively by contraction-deletion rules. Well known examples are the Tutte, chromatic, tension and flow polynomials. Suppose that an edge cut C divides a graph G into two parts G1, G1′ and that G1, G1′ are the sets of minors of G whose edge sets consist of C and edges of G1, G1′, respectively. We study determinant formulas evaluating a Tutte-Grothendieck invariant of G from the Tutte-Grothendieck invariants of graphs from G1 and G1′.