Testing Full Outer-2-planarity in Linear Time
A graph is 1-planar, if it admits a 1-planar embedding, where each edge has at most one crossing. Unfortunately, testing the 1-planarity of a graph is known as NP-complete.
This paper initiates the study of the problem of the testing 2-planarity of a graph, in particular, testing the “full-outer-2-planarity” of a graph. A graph is outer-2-planar, if it admits an outer-2-planar embedding, that is every vertex is on the outer boundary and no edge has more than two crossings. A graph is fully-outer-2-planar, if it admits a fully-outer-2-planar embedding, that is an outer-2-planar embedding such that no crossing appears along the outer boundary. We present several structural properties of triconnected outer-2-planar graphs and fully-outer-2-planar graphs, and prove that triconnected fully-outer-2-planar graphs have a constant number of fully-outer-2-planar embeddings. Based on these properties, we present linear-time algorithms for testing the fully-outer-2-planarity of a graph G, whose vertex-connectivity is 1, 2 or at least 3. The algorithm also produces a fully-outer-2-planar embedding of a graph, if it exists. Moreover, we show that every fully-outer-2-planar embedding admits a straight-line drawing.
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