A linear time algorithm for testing maximal 1-planarity of graphs with a rotation system

Authors:
Peter Eades;Seok-Hee Hong;Naoki Katoh;Giuseppe Liotta;Pascal Schweitzer;Yusuke Suzuki
Affiliations:
University of Sydney, Australia;University of Sydney, Australia;Kyoto University, Japan;Universití di Perugia, Italy;Australian National University, Australia;Niigata University, Japan
Venue:
Theoretical Computer Science
Year:
2013

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Abstract

A 1-planar graph is a graph that can be embedded in the plane with at most one crossing per edge. It is known that testing 1-planarity of a graph is NP-complete. In this paper, we consider maximal 1-planar graphs. A graph G is maximal 1-planar if addition of any edge destroys 1-planarity of G. We first study combinatorial properties of maximal 1-planar embeddings. In particular, we show that in a maximal 1-planar embedding, the graph induced by the non-crossing edges is spanning and biconnected. Using the properties, we show that the problem of testing maximal 1-planarity of a graph G can be solved in linear time, if a rotation system @F (i.e., the circular ordering of edges for each vertex) is given. We also prove that there is at most one maximal 1-planar embedding @x of G that is consistent with the given rotation system @F. Our algorithm also produces such an embedding in linear time, if it exists.