We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., \emph{convex drawings}. We consider two families of graph classes with convex drawings: \emph{outer $k$-planar} graphs, where each edge is crossed by at most $k$ other edges; and \emph{outer $k$-quasi-planar} graphs, where no $k$ edges can mutually cross. We show that the outer $k$-planar graphs are $\lfloor3.5\sqrt{k}\rfloor$-degenerate, and consequently that every outer $k$-planar graph can be colored with $\lfloor3.5\sqrt{k}\rfloor + 1$ colors. We further show that every outer $k$-planar graph has a balanced vertex 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, e.g., this implies that none of these recognition problems is NP-hard unless the Exponential Time Hypothesis fails. We also show that the class of outer 3-quasi-planar graphs and the class of planar graphs are incomparable. Finally, we restrict outer $k$-planar and outer $k$-quasi-planar drawings to \emph{full} drawings (where no crossing appears on the boundary of the outer face) and to \emph{closed} drawings (where the vertex sequence on the boundary of the outer face is a Hamiltonian cycle in the graph). For each $k$, we express \emph{closed outer $k$-planarity} and \emph{closed outer $k$-quasi-planarity} in extended monadic second-order logic. Since every outer $k$-planar graph has treewidth $O(k)$, Courcelle's theorem implies that closed outer $k$-planarity is linear-time testable. We leverage this result to further show that full outer $k$-planarity can also be tested in linear time.

翻译：我们研究图中顶点在平面上呈凸位置排列的直线画法，即凸画法。考虑两类具有凸画法的图类：外$k$平面图（每条边至多被其他$k$条边交叉）和外$k$拟平面图（不存在$k$条边相互交叉）。我们证明外$k$平面图是$\lfloor3.5\sqrt{k}\rfloor$退化的，因此每个外$k$平面图可用$\lfloor3.5\sqrt{k}\rfloor + 1$种颜色着色。进一步证明每个外$k$平面图存在大小不超过$2k+3$的平衡顶点分隔符。对于每个固定$k$，这些小的平衡分隔符使我们能在拟多项式时间内检验外$k$平面性，这意味着除非指数时间假说失败，否则这些识别问题均非NP难问题。我们还证明外3拟平面图类与平面图类不可比较。最后，我们将外$k$平面图和外$k$拟平面图画法限制为完全画法（外边界无交叉）和封闭画法（外边界顶点序列为图的哈密顿环）。对每个$k$，我们用扩展的一元二阶逻辑表达封闭外$k$平面性和封闭外$k$拟平面性。由于每个外$k$平面图的树宽为$O(k)$，库塞尔定理表明封闭外$k$平面性可在线性时间内检验。我们进一步利用此结果证明完全外$k$平面性也可在线性时间内检验。