Simple Realizability of Complete Abstract Topological Graphs Simplified

  • Jan KynčlEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)


An abstract topological graph (briefly an AT-graph) is a pair \(A=(G,\mathcal {X})\) where \(G=(V,E)\) is a graph and \(\mathcal {X}\subseteq \left( {\begin{array}{c}E\\ 2\end{array}}\right) \) is a set of pairs 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 \(\mathcal {X}\) crosses exactly once and no other pair crosses. We characterize simply realizable complete AT-graphs 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.



I thank Martin Balko for his comments on an earlier version of the manuscript. I also thank all the reviewers for their suggestions for improving the presentation.


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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Institute for Theoretical Computer Science, Faculty of Mathematics and PhysicsCharles UniversityPraha 1Czech Republic
  2. 2.Chair of Combinatorial Geometry, EPFL-SB-MATHGEOM-DCGÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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