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−(22k−1). 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 \emph{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 \emph{closed outer k-quasi-planarity} in extended monadic
second-order logic. Thus, closed outer k-planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017