Extension of deletion–constriction and current linearity to arbitrary subsets of edges

Determine whether the second-order deletion–constriction relations for products of rooted spanning-tree polynomials and the resulting mutual linearity of stationary Markov currents, established for two pinned edges, extend to arbitrary subsets of edges in weighted directed graphs.

Background

Using tree-surgery techniques, the paper derives a second-order deletion–constriction formula for products of rooted spanning-tree polynomials that leads to coplanarity and mutual linearity of currents for the case of one pinned edge, and extends these ideas to two edges under technical conditions.

The authors conjecture that analogous structural results—both in terms of spanning-tree vector geometry and induced current linearity—may hold for arbitrary subsets of edges beyond the two-edge case. Establishing such generalizations would broaden the applicability of the framework to multi-edge control scenarios in Markov processes.