Beyond Outerplanarity

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 \((\lfloor \sqrt{4k+1}\rfloor +1)\)-degenerate, and consequently that every outer k-planar graph can be \((\lfloor \sqrt{4k+1}\rfloor +2)\)-colored, and this bound is tight. We further show that every outer k-planar graph has a balanced separator of size at most \(2k+3\). For each fixed k, these small balanced separators allow us to test outer k-planarity in quasi-polynomial time, i.e., none of these recognition problems are np-hard unless eth fails.for the outer k-quasi-planar graphs we discuss the edge-maximal graphs which have been considered previously under different names. 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, since outer k-planar graphs have bounded treewidth, closed outer k-planarity is linear-time testable by courcelle’s theorem.