Comments on What is an example of a pathological imbedding of a(n allowed) graph into an oriented surface?
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What is an example of a pathological imbedding of a(n allowed) graph into an oriented surface?
I am reading the following paper: G. A. Jones, and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), no. 2, 273–307 (MR0505721, Zbl 0391.05024). The authors are interested in imbeddings of an allowed graph
there is a homeomorphism of
with a subspace of [and] we will identify with its image in …
Here, an "allowed graph" is slightly more general than an undirected pseudograph in that it allows the existence of edges that are free at one end; e.g., let
Now, on page 277, the authors make an additional assumption about the imbeddings of allowed graphs into oriented surfaces:
TM1: whenever
has valency , then there is a neighborhood of in and a homeomorphism such that and . This condition is imposed to avoid pathological imbeddings.
Question: What is an example of a pathological imbedding of an allowed graph, i.e., one that violates the assumption TM1?
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