Peripheral circuits in infinite binary matroids

Abstract

A peripheral circuit in a (possibly infinite) matroid M is a circuit C of M such that M/C is connected. In the case of a 3-connected graph G, this is equivalent to C being chordless and G−V(C) being connected. Two classical theorems of Tutte assert that, for a 3-connected graph G: (i) every edge e of G is in two peripheral cycles that intersect just on e and its incident vertices; and (ii) the peripheral cycles generate the cycle space of G[12].Bixby and Cunningham generalized these to binary matroids, with (i) requiring a small adaptation. Bruhn generalized (i) and (ii) to the Freudenthal compactification of a locally finite graph. We unify these two generalizations to “cofinitary, binary B-matroids”. (Higgs introduced the B-matroid as an infinite matroid; recent works show this should now be accepted as the right notion of infinite matroid. Cofinitary means every cocircuit is finite.)