Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: The Planarity Theorems of MacLane and Whitney for Graph-like Continua. | The Planarity Theorems of MacLane and Whitney for Graph-like Continua Robin Christian R. Bruce Richter Brendan Rooney Department of Combinatorics and Optimization University of Waterloo Waterloo Canada N2L 3G1 Submitted Mar 12 2009 Accepted Dec 11 2009 Published Jan 5 2010 Mathematics Subject Classification 05C10 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 . 1 Introduction The theorems of MacLane and Whitney characterize planarity of a graph G in different ways. The former characterizes planar graphs by the existence of a basis for the cycle space of G in which every edge appears at most twice while the latter characterization is in terms of the circuit matroid of G having a graphic dual matroid. Naturally these are closely connected to Kuratowski s Theorem G is planar if and only if G does not contain a subdivision of either K3 3 or K5. Recent work has treated extensions of these theorems to infinite graphs. Bruhn and Diestel 1 generalize Whitney s Theorem in the case of an infinite graph in which no two vertices are joined by infinitely many edge-disjoint paths. Bruhn and Stein 2 provide MacLane s Theorem in the case of locally finite graphs. In both cases the theorem is proved for a topological space obtained from the graph by adding its ends and in the former case identifying a vertex with each end that is not separated from the vertex by any finite set of edges. In the central case of 2-connected graphs these spaces are compact connected and Hausdorff. Thomassen and Vella 12 have recently introduced the notion of a graph-like space. A graph-like space is a metric space G that contains a subset V so that i for any
