The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer $k$-planar graphs, that is, graphs that admit a straight-line drawing where all the vertices lie on a circle, and every edge is crossed by at most $k$ other edges. Wood and Telle [New York J. Math., 2007] showed that every outer $k$-planar graph has treewidth at most $3k + 11$ using so-called planar decompositions, and later, Auer et al. [Algorithmica, 2016] proved that the treewidth of outer $1$-planar graphs is at most $3$, which is tight. In this paper, we improve the general upper bound to $1.5k + 2$ and give a tight bound of $4$ for $k = 2$. We also establish a lower bound: we show that, for every even $k$, there is an outer $k$-planar graph with treewidth $k+2$. Our new bound immediately implies a better bound on the cop number, which answers an open question of Durocher et al. [GD 2023] in the affirmative. Our treewidth bound relies on a new and simple triangulation method for outer $k$-planar graphs that yields few crossings with graph edges per edge of the triangulation. Our method also enables us to obtain a tight upper bound of $k + 2$ for the separation number of outer $k$-planar graphs, improving an upper bound of $2k + 3$ by Chaplick et al. [GD 2017]. We also consider outer min-$k$-planar graphs, a generalization of outer $k$-planar graphs, where we achieve smaller improvements.

翻译：树宽是衡量图结构树相似性的参数。许多算法与组合结果均以树宽表述。本文研究外 $k$-平面图的树宽，即存在直线绘制方式使得所有顶点位于同一圆周上，且每条边至多被其他 $k$ 条边交叉的图。Wood 与 Telle [New York J. Math., 2007] 通过平面分解方法证明所有外 $k$-平面图的树宽至多为 $3k + 11$；随后 Auer 等人 [Algorithmica, 2016] 证明外 $1$-平面图的树宽至多为 $3$（该界是紧的）。本文中，我们将一般上界改进至 $1.5k + 2$，并针对 $k = 2$ 的情形给出紧上界 $4$。同时我们建立了下界：证明对任意偶数 $k$，存在树宽为 $k+2$ 的外 $k$-平面图。我们的新界立即导出警察数更优的上界，这肯定地回答了 Durocher 等人 [GD 2023] 提出的开放问题。我们的树宽界依赖于一种新颖且简洁的外 $k$-平面图三角剖分方法，该方法使得三角剖分中每条边与图边的交叉数较少。此方法还使我们能为外 $k$-平面图的分离数获得紧上界 $k + 2$，改进了 Chaplick 等人 [GD 2017] 提出的 $2k + 3$ 上界。我们还考虑了外最小 $k$-平面图（外 $k$-平面图的推广形式），在此类图中我们取得了相对较小的改进。