Beyond Outerplanarity
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We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., convex drawings. We consider two families of graph classes with nice convex drawings: outer k-planar graphs, where each edge is crossed by at most k other edges; and, outer k-quasi-planar graphs where no k edges can mutually cross. We show that the outer k-planar graphs are (√(4k+1)+1)-degenerate, and consequently that every outer k-planar graph can be (√(4k+1)+2)-colored, and this bound is tight. We further show that every outer k-planar graph has a balanced separator of size O(k). This implies that every outer k-planar graph has treewidth O(k). For fixed k, these small balanced separators allow us to obtain a simple quasi-polynomial time algorithm to test whether a given graph is outer k-planar, i.e., none of these recognition problems are NP-complete unless ETH fails. For the outer k-quasi-planar graphs we prove that, unlike other beyond-planar graph classes, every edge-maximal n-vertex outer k-quasi planar graph has the same number of edges, namely 2(k-1)n - 2k-12. We also construct planar 3-trees that are not outer 3-quasi-planar. Finally, we restrict outer k-planar and outer k-quasi-planar drawings to closed drawings, where the vertex sequence on the boundary is a cycle in the graph. For each k, we express closed outer k-planarity and closed outer k-quasi-planarity in extended monadic second-order logic. Thus, closed outer k-planarity is linear-time testable by Courcelle's Theorem.
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