Beyond Outerplanarity

Abstract

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 kplanar 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.

A Survey on Graph Drawing Beyond Planarity

“…Convex geometric k-planar graphs in which all the vertices form a simple cycle can be recognized in linear time, because they can be expressed in extended monadic second-order logic and have bounded threewidth. In [73] we can also find families of 3-quasi planar graphs that are convex geometric and families that are not convex geometric.…”

“…Note that, the edge crossings in a geometric graph with all vertices in convex position are the same as the edge crossings of a 1-page drawing in which the order of the vertices along the spine is the same as the circular order of the vertices on the convex polygon. Several interesting properties of convex geometric k-planar and k-quasi planar graphs have been recently investigated [73]. It is shown that convex geometric k-planar graphs are √ 4k + 1 +1-degenerate and consequently √ 4k + 1 + 2-colorable.…”

“…Note that, the edge crossings in a geometric graph with all vertices in convex position are the same as the edge crossings of a 1-page drawing in which the order of the vertices along the spine is the same as the circular order of the vertices on the convex polygon. 2-connected 2-layer fan-planar O(n) [47] outer 1-planar O(n) [26,141] full outer 2-planar O(n) [143] maximal outer fan-planar O(n) [39] circular RAC O(n) [84] upward RAC NP-hard [17] simultaneous RAC NP-hard [134] k-SEFE NP-complete [134] straight-line point-set RAC NP-hard [128] Several interesting properties of convex geometric k-planar and k-quasi planar graphs have been recently investigated [73]. It is shown that convex geometric k-planar graphs are √ 4k + 1 +1-degenerate and consequently √ 4k + 1 + 2-colorable.…”

Edge-Minimum Saturated k-Planar Drawings

“…Seemingly the only min-saturation results involving this crossing restriction concern so-called outer or convex drawings in which all of the vertices occur on the boundary of a single face of the drawing. Here, as in planar drawings, the min-saturated drawings and max-saturated outer k-quasiplanar n-vertex drawings coincide [12,14,25], and have 2(k − 1)n − 2k−1 2 edges [11]. Also for the concept of gap-planarity [7], which generalizes the notion of k-planarity, the focus so far has been on the Turán-type question.…”

Bundled Crossings Revisited

“…To prove our main result (Theorem 2) we develop an algorithm that tests whether bc • (G) = k in FPT time with respect to k. Our algorithm is inspired by recent works on circular layouts with at most k crossings [3] and circular layouts where each edge is crossed at most k times [4]. In both of these prior works, it is first observed that the graphs admitting such circular layouts have treewidth O(k), and then algorithms are developed using Courcelle's theorem, which establishes that expressions in MSO 2 logic can be evaluated efficiently.…”