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Abstract
We relate matroid connectivity to Tutte-connectivity in an infinite graph. Moreover, we show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which also infinite cycles are taken into account, have the same connectivity function. As an application we re-prove that, also for infinite graphs, Tutte-connectivity is invariant under taking dual graphs.
Highlights
We show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which infinite cycles are taken into account, have the same connectivity function
This work is part of a project to develop a theory for infinite matroids that is analogous to its finite counterpart
The main result of this work is an extension of this fact to infinite graphs and matroids
Summary
This work is part of a project to develop a theory for infinite matroids that is analogous to its finite counterpart. A finite matroid M is k-connected if for any k and any partition of its ground set into two sets X, Y of at least elements each it follows that r(X) + r(Y ) − r(M ) This definition is useless for infinite matroids as the involved ranks will usually be infinite. An infinite cycle in the graph is the homeomorphic image of the unit circle in a natural topological space obtained from the graph (often by compactifying it) This definition was proposed by Diestel and Kuhn in a completely graph-theoretical context and was subsequently seen to be extremely fruitful as it allows to extend virtually any result about cycles in a finite graph to at least a large class of infinite graphs; see Diestel [11] for an introduction.
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