Recognizing Outer 1-Planar Graphs in Linear Time

Auer, Christopher and Bachmaier, Christian and Brandenburg, Franz J. and Gleißner, Andreas and Hanauer, Kathrin and Neuwirth, Daniel and Reislhuber, Josef (2013) Recognizing Outer 1-Planar Graphs in Linear Time. In: 21st International Symposium, GD 2013, September 23-25, 2013 , pp. 107-118(Official URL:

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A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are on the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time and specialize 1-planar graphs, whose recognition is NP -hard. Our main result is a linear-time algorithm that first tests whether a graph G is o1p, and then computes an embedding. Moreover, the algorithm can augment G to a maximal o1p graph. If G is not o1p, then it includes one of six minors (see Fig. 3), which are also detected by the recognition algorithm. Hence, the algorithm returns a positive or negative witness for o1p.

Item Type: Conference Paper
Classifications: G Algorithms and Complexity > G.420 Crossings
G Algorithms and Complexity > G.490 Embeddings
G Algorithms and Complexity > G.770 Planarity Testing

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