Infinite Graphic Matroids

Abstract

We generalise the construction of infinite matroids from trees of matroids to allow the matroids at the nodes, as well as the field over which they are represented, to be infinite.

Graph‐like compacta: Characterizations and Eulerian loops

“…Then (2) is satisfied since vertices of F 0 are mid-points of edges of G 0 , and every open edge of F 0 is disjoint from G 0 ; and (1) is trivially true. Now inductively, suppose we have already defined H n = G n ∪ F n for some n ∈ N according to (1) and (2). First, consider the natural pull-backF n ⊂ G n+1 of F n under f n .…”

“…First, consider the natural pull-backF n ⊂ G n+1 of F n under f n . More precisely, by (2), the preimage f −1 n (F n ) ⊂ G n+1 is isomorphic to a subgraph of F n . LetF n be an isomorphic copy of F n on the vertex set f −1 n (V (F n )) obtained by adding all edges missing from f −1 n (F n ) so that they are disjoint from G n+1 .…”

“…Papers in which graph-like spaces have played a key role include [22] where several Menger-like results are given, and [8] where algebraic criteria for the planarity of graph-like continua are presented. In [2], aspects of the matroid theory for graphs have been generalized to infinite matroids on graph-like spaces.…”

n-arc and n-circle connected graph-like spaces

“…is n-cc but not (n + 1)-cc. Let us write [v k ] ∈ Z for the vertex corresponding to the equivalence class of v (1) k . Then it is clear from the construction that deleting S = {[v 1 ], .…”

Topological cycle matroids of infinite graphs

“…We shall also need the following related lemma, which is a combination of Lemma 6.6 and Lemma 6.8 from [4].…”