Graph and map isomorphism and all polyhedral embeddings in linear time

For every surface S (orientable or non-orientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of face-width at least 3 into S. This improves a previously known algorithm whose time complexity is nO(g), where g is the genus of S. This is the first algorithm for which the degree of polynomial in the time complexity does not depend on g. The above result is based on two linear time algorithms, each of which solves a problem that is of independent interest. The first of these problems is the following one. Let S be a fixed surface. Given a graph G and an integer k ≥ 3, we want to find an embedding of G in S of face-width at least k, or conclude that such an embedding does not exist. It is known that this problem is NP-hard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of face-width at least k, up to Whitney equivalence. Here, the face-width of an embedded graph G is the minimum number of points of G in which some non-contractible closed curve in the surface intersects the graph. In the proof of the above algorithm, we give a simpler proof and a better bound for the theorem by Mohar and Robertson concerning the number of polyhedral embeddings of 3-connected graphs. The second ingredient is a linear time algorithm for map isomorphism and Whitney equivalence. This part generalizes the seminal result of Hopcroft and Wong that graph isomorphism can be decided in linear time for planar graphs.

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