Cycle Matroids of Graphings: From Convergence to Duality

TL;DR

This paper explores the connection between graph convergence concepts and matroid theory, specifically analyzing cycle matroids of graphings, their duality, and the conditions under which these properties hold.

Contribution

It introduces a new convergence notion for matroids based on graph limits and characterizes duality conditions for cycle and cocycle matroids in graphings.

Findings

Characterizes exposed points of convex sets related to matroid polytopes

Establishes a link between local-global graph convergence and matroid quotient-convergence

Shows duality of cycle and cocycle matroids in hyperfinite planar graphings

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. Motivated by the limit theory of graphs, the authors introduced a form of right-convergence, called quotient-convergence, for a sequence of submodular setfunctions, leading to a notion of convergence for matroids through their rank functions. In this paper, we study the connection between local-global convergence of graphs and quotient-convergence 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.