Lecture Notes in Computer Science Volume 8242, 2013, pp 71-82

A Linear-Time Algorithm for Testing Outer-1-Planarity

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A graph is 1-planar if it can be embedded in the plane with at most one crossing per edge. A graph is outer-1-planar if it has an embedding in which every vertex is on the outer face and each edge has at most one crossing. We present a linear time algorithm to test whether a graph is outer-1-planar. The algorithm can be used to produce an outer-1-planar embedding in linear time if it exists.

This paper is an extended abstract. For omitted proofs, see the full version of this paper [1]. The problem studied in this paper was initiated at the Port Douglas Workshop on Geometric Graph Theory, June, 2011, held in Australia, organized by Peter Eades and Seok-Hee Hong, supported by IPDF funding from the University of Sydney.
Independently, another linear time algorithm is reported in [2].