Embedding Graphs in the Torus in Linear Time

Let K be a subgraph of G, and suppose that we are given a (2-cell) embedding of K into a surface S. The embedding extension problem asks whether it is possible to extend the given embedding of K to an embedding of G in S . Every such embedding is called an embedding extension of K to G. An obstruction for embedding extensions is a subgraph 12 of G E ( K ) such that the embedding of K cannot be extended to KUI2. The obstruction is small if KUI2 is homeomorphic to a graph with a small number of edges. If [2 is small, then it is easy to verify (for example, by checking all the possibilities for the rotation systems of K U 12) that no embedding extension to h" U f2 exists, and hence ~ is a good verifier that there are no embedding extensions of K to G as well. Though obstructions can be arbitrarily large, one can produce a small obstruction by changing some branches of K. Such changes are often applied in our algorithms. To indicate that the graph K might have been changed, we call a small obstruction obtained in this way a n'~ce obstruction. It is known [15] that the general problem of determining the genus, or the non-orientable genus of graphs is NP-hard. However, for every fixed surface there is a polynomial time algorithm which checks if a given graph can be embedded in the surface. Such algorithms were found first by Filotti et al. [3]. Unfortunately, even for the torus their algorithm has time complexity estimated only by O(nlSS). A special polynomial time algorithm for embedding cubic graphs in the torus has been published by Filotti [2]. Robertson and Seymour developed an O(n 3) algorithm using graph minors (with recent improvement by B. Reed to O(n21ogn))[12, 13, 14]. The main theorem of the present paper is: