Abstract
An abstract topological graph (briefly an AT-graph) is a pair A = (G,R) where G = (V,E) is a graph and \(R\subseteq {E \choose 2}\) is a set of pairs of its edges. An AT-graph A is simply realizable if G can be drawn in the plane in such a way that each pair of edges from R crosses exactly once and no other pair crosses. We present a polynomial algorithm which decides whether a given complete AT-graph is simply realizable. On the other hand, we show that other similar realizability problems for (complete) AT-graphs are NP-hard.KeywordsRealizability ProblemRotation SystemIsomorphism ClassTopological GraphCyclic OrderThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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