Graphs Embedded in the Plane with Finitely Many Accumulation Points

C. Paul Bonnington and R. Bruce Richter

Abstract

Halin's Theorem characterizes those infinite connected
graphs that have an embedding in
the plane with no accumulation points, by exhibiting the list of
excluded subgraphs. We generalize this by obtaining a similar
characterization of which infinite connected graphs have an embedding
in the plane (and other surfaces) with at most $k$ accumulation points.
Thomassen~cite{T} provided a different characterization of those infinite
connected graphs that have an embedding in the plane with no
accumulation points as those for which the ${bf Z}_2$-vector space
generated by the cycles has a basis for which every edge is in at most
two members. Adopting the definition that the cycle space is the set
of all edge-sets of subgraphs in which every vertex has even degree
(and allowing restricted infinite sums),
we prove a general analogue of
Thomassen's result, obtaining a cycle space characterization of a graph
having an embedding in the sphere with $k$ accumulation points.


Keywords
Infinite planar graphs, accumulation points

Math Review Classification
Primary 05C10

Last Updated

Length
13

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