Abstract

There exist very efficient algorithms to decide whether a graph is planar. How difficult can it be to decide whether a graph is embeddable in some specified subset S of the plane? It is shown that the time complexity of such embeddability problems can be prescribed in the following sense. To every class C of graphs one can associate a path-connected subset p( C) of the plane, defined as a topological realization of an appropriate countable graph, in such a way that the problem of recognizing the class C is polynomial time equivalent to the problem of testing embeddability in p( C). Incidentally, a similar result is proved on combinatorial embeddability in infinite graphs.

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