Embedding a graph-like continuum in a surface

Abstract

Graph-like continua provide a very natural setting for generalizing finite graphs to infinite, compact structures. For example, the Freudenthal compactification of a locally finite graph, exploited by Diestel and his students in their study of the cycle space of an infinite graph, is an example of a graph-like continuum. Generalizing earlier works in special cases, the authors, along with Christian, have proved MacLane’s and Whitney’s characterizations of planarity for graph-like continua (Electron. J. Combin. 17 (2010)). In this work, we consider embeddings of graph-like continua in compact surfaces and show that: (i) every edge is in an open disc that meets the graph-like continuum precisely in that edge; (ii) there are natural analogues of face boundary walks; (iii) there is a graph-like continuum triangulating the same surface and containing as a sub-graphlike continuum the original embedded graph-like continuum; (iv) the face boundaries generate a subspace of the cycle space; and (v) the quotient of the cycle space by the boundary cycles is the homology of the surface. These all generalize results known for embeddings of finite graphs.