Simple realizability of complete abstract topological graphs simplified

Abstract

An abstract topological graph (briefly an(A)-graph) is a pair A = (G,X) where G = (V,E) is a graph and X ⊆ E is a set of pairs 2 of its edges. The AT-graph A is simply realizable if G can be drawn in the plane so that each pair of edges from X crosses exactly once and no other pair crosses. We characterize simply realizable complete ATgraphs by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author.