Testing Maximal 1-Planarity of Graphs with a Rotation System in Linear Time - (Extended Abstract)

A 1-planar graph is a graph that can be embedded in the plane with at most one crossing per edge. A 1-planar graph on n vertices can have at most 4n− 8 edges. It is known that testing 1-planarity of a graph is NP-complete. A 1-planar embedding of a graph G is maximal, if no edge can be added without violating the 1-planarity of G. In this paper, we study combinatorial properties of maximal 1-planar embeddings. In particular, we show that in a maximal 1planar 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 Φ (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 ξ ofG that preserves the given rotation system Φ, and our algorithm also produces such an embedding in linear time, if it exists. In addition, we establish new bounds on the minimum number of edges of maximal 1-plane graphs, showing that for graphs on n vertices this number is between 9n 5 − 18 5 and 7n 3 − 2.