An obstacle representation of a graph $G$ consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of $G$ to points such that two vertices are adjacent in $G$ if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each $n$-vertex graph is $O(n \log n)$ [Balko, Cibulka, and Valtr, 2018] and that there are $n$-vertex graphs whose obstacle number is $\Omega(n/(\log\log n)^2)$ [Dujmovi\'c and Morin, 2015]. We improve this lower bound to $\Omega(n/\log\log n)$ for simple polygons and to $\Omega(n)$ for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of $n$-vertex graphs with bounded obstacle number, solving a conjecture by Dujmovi\'c and Morin. We also show that if the drawing of some $n$-vertex graph is given as part of the input, then for some drawings $\Omega(n^2)$ obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph $G$ is fixed-parameter tractable in the vertex cover number of $G$. Second, we show that, given a graph $G$ and a simple polygon $P$, it is NP-hard to decide whether $G$ admits an obstacle representation using $P$ as the only obstacle.

翻译：图 $G$ 的一个障碍表示包含一组两两不相交的单连通闭区域，以及将 $G$ 的顶点一一映射到点集，使得 $G$ 中两个顶点相邻当且仅当连接这两个对应点的线段不与任何障碍相交。图的障碍数是其在平面上所有障碍均为简单多边形时，障碍表示中所用障碍的最小数目。已知每个 $n$ 顶点图的障碍数为 $O(n \log n)$ [Balko, Cibulka, and Valtr, 2018]，且存在 $n$ 顶点图的障碍数为 $\Omega(n/(\log\log n)^2)$ [Dujmovi\'c and Morin, 2015]。我们将简单多边形的下界改进为 $\Omega(n/\log\log n)$，将凸多边形的下界改进为 $\Omega(n)$。为获得这些更强的界，我们改进了具有有界障碍数的 $n$ 顶点图数量的已知估计，解决了 Dujmovi\'c 和 Morin 的一个猜想。我们还证明，若某些 $n$ 顶点图的画法作为输入给定，则对于某些画法需要 $\Omega(n^2)$ 个障碍才能将其转化为该图的障碍表示。我们的界在若干情形下是渐近紧的。我们通过两个复杂性结果补充这些组合界。首先，我们证明计算图 $G$ 的障碍数在 $G$ 的顶点覆盖数参数下是固定参数可解的。其次，我们证明，给定图 $G$ 和简单多边形 $P$，判定 $G$ 是否允许以 $P$ 为唯一障碍的障碍表示是 NP 困难的。