The Planarity Theorems of MacLane and Whitney for Graph-like Continua

Abstract

The planarity theorems of MacLane and Whitney are extended to compact graph-like spaces. This generalizes recent results of Bruhn and Stein (MacLane's Theorem for the Freudenthal compactification of a locally finite graph) and of Bruhn and Diestel (Whitney's Theorem for an identification space obtained from a graph in which no two vertices are joined by infinitely many edge-disjoint paths).

On graph-like continua of finite length

“…In both cases, J is contained in Q, and hence in I. But as d ⊆ J by definition, this contradicts the assumption that d\I is non-empty, proving that I is the subarc sought after by (5).…”

“…Since for every q ∈ ∂K we have q ′ ∈ B by the definition of B, (5) implies that q ′ lies in a maximal super-bridged subarc P q for every such q (we might have P q = P r for q = r ∈ ∂K though). Similarly, each q ∈ {P 0 , P 1 }, there is a maximal super-bridged subarc P q containing q.…”

“…A lot of recent work is devoted to extending fundamental theorems from finite graphs to infinite ones using topology; [6] gives a survey of some 40 papers of this kind. Some further research is emerging that uses this experience in order to extend such theorems to general topological spaces, usually continua [8,5,25]. In this context, Thomassen and Vella [25] introduce the notion of a graph-like space, defined as a topological space X containing a set E of pairwise disjoint copies of R, called edges, each of which is open in X and has exactly two points in its frontier, such that the subspace X\ E is totally disconnected.…”

“…Several well-known graph-theoretic results have already been extended to graph-like continua [5,25]. In this paper we develop tools for studying analytic properties of these spaces.…”

Graph‐like compacta: Characterizations and Eulerian loops

“…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.…”