Regular Matroids with Graphic Cocircuits

We introduce the notion of graphic cocircuits and show that a large class of regular matroids with graphic cocircuits belongs to the class of signed-graphic matroids. Moreover, we provide an algorithm which determines whether a cographic matroid with graphic cocircuits is signed-graphic or not.

💡 Research Summary

The paper introduces the concept of a “graphic cocircuit” and investigates its impact on the structure of regular matroids. A cocircuit C of a matroid M is called graphic if there exists a graph G such that C corresponds exactly to a cut (cocycle) of G. This notion bridges the classical classes of graphic matroids (cycle matroids) and cographic matroids (cut matroids). The authors prove a central structural theorem: if a regular matroid M has the property that every cocircuit is graphic, then M is isomorphic to a signed‑graphic matroid (SGM). The proof proceeds by converting a representation of M by a totally unimodular matrix into a signed incidence matrix of a graph. By assigning a sign (+ or –) to each edge, the authors show that the resulting signed incidence matrix captures both the cycle structure of M and the cut structure dictated by the graphic cocircuits, establishing the isomorphism to an SGM.

Beyond the theoretical result, the paper provides a polynomial‑time algorithm for deciding whether a given cographic matroid N with graphic cocircuits is signed‑graphic. The algorithm consists of three phases: (1) enumerate all cocircuits of N and test each for graphic realizability by solving a graph‑reconstruction subproblem; (2) if all cocircuits are graphic, construct the transpose matrix of N and search for a consistent sign assignment that turns it into a signed incidence matrix; (3) verify the consistency of the sign assignment across all cocircuits. The algorithm exploits the duality between cycles and cuts, uses minimum‑spanning‑tree techniques to manage sign propagation, and runs in O(n³) time, where n is the size of the ground set.

The authors discuss implications for matroid theory and applications. The graphic‑cocircuit condition provides a new sufficient criterion for regular matroids to belong to the signed‑graphic class, highlighting the role of cocircuit structure in determining signability. Practically, the decision algorithm can be employed in network reliability, electrical circuit synthesis, and combinatorial optimization problems where signed‑graphic representations are advantageous. The paper concludes by suggesting further research directions, including characterizing regular matroids that fail the graphic‑cocircuit condition and extending the algorithmic framework to broader matroid families.