Cycle matroids of graphings: From convergence to duality

Bérczi, Kristóf and Borbényi, Márton and Lovász, László and Tóth, László Márton (2026) Cycle matroids of graphings: From convergence to duality. JOURNAL OF COMBINATORIAL THEORY SERIES B, 178. pp. 118-144. ISSN 0095-8956

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Abstract

A recent line of research has concentrated on exploring the links between analytic and combinatorial theories of submodularity, uncovering several key connections between them. In this context, Lov´asz initiated the study of matroids from an analytic point of view and introduced the cycle matroid of a graphing [23]. Motivated by the limit theory of graphs, the authors introduced a form of rightconvergence, called quotient-convergence, for a sequence of submodular setfunctions, leading to a notion of convergence for matroids through their rank functions [5]. In this paper, we study the connection between local-global convergence of graphs and quotientconvergence of their cycle matroids. We characterize the exposed points of associated convex sets, forming an analytic counterpart of matroid independence- and base-polytopes. Finally, we consider dual planar graphings and show that the cycle matroid of one is the cocycle matroid of its dual if and only if the underlying graphings are hyperfinite.

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