Bounding and computing obstacle numbers of graphs

TL;DR

This work advances the theory of obstacle representations of graphs by tightening lower bounds on obstacle numbers for general and convex obstacles, improving the combinatorial bounds on the number of graphs with bounded obstacle number, and establishing notable algorithmic results. The authors prove $ ext{obs}(n)\in ext{Ω}(n/ ext{log} ext{log} n)$ for simple polygons and $ ext{obs}_{c}(n)\in ext{Ω}(n)$ for convex polygons, deriving these via strengthened upper bounds on $f(h,n)$ and $f_c(h,n)$, respectively. They show $f(h,n)\in 2^{O(hn ext{log} n)}$ and $f_c(h,n)\in 2^{O(n(h+ ext{log} n))}$, and prove a quadratic lower bound on obstacle numbers of some graph drawings, plus an asymptotically tight bound $ ext{max}_D ext{obs}(D)\in ext{Ω}(n^{2})$ for drawings. Algorithmically, the obstacle number problem is FPT when parameterized by vertex cover size, and deciding representation with a given polygon is NP-hard. These results collectively narrow the gap between upper and lower bounds, and illuminate the computational complexity landscape of obstacle representations.

Abstract

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 $Ω(n/(\log\log n)^2)$ [Dujmović and Morin, 2015]. We improve this lower bound to $Ω(n/\log\log n)$ for simple polygons and to $Ω(n)$ 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 Dujmović 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 $Ω(n^2)$ 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.

Figures (14)

• Figure 1: (a) An illustration of the statement of Lemma \ref{['lem-boundaryIntersections']}. (b) An example where $u$ is responsible for $t$ and $w$ but not for $v$. The vertex $u'$ is responsible for $v$.
• Figure 2: (a) An illustration of the proof of Lemma \ref{['lem-responsbility']}. (b) The reflex vertex $w$ is responsible for the non-blocking convex vertices $v$ and $v'$ of the obstacle $O$.
• Figure 3: An illustration of the proof of Lemma \ref{['lem-minimalObstaclesReflex']}. (a) Merging two obstacles. (b) If $w \in O$, then there is an isolated component $(D'\mathcal{O}')$ in the region $C$ (denoted by stripes).
• Figure 4: An illustration of the proof of Lemma \ref{['lem-minimalObstaclesReflex']}. Reducing the number of vertices of an obstacle $O$ by filling up regions $C'_1,\dots,C'_5$ and "drilling" a vertical tunnel from the vertex $t$ of $C_6$.
• Figure 5: An example of a vertex set $V$ with its dual line arrangement $\mathcal{A}= V^*$. Points and lines of the same color are dual to each other. The curve $\tau^*$ for the upper envelope $O^+$ of the obstacle $O$ is denoted black and the curve $\beta^*$ for the lower envelope $O^-$ of the obstacle $O$ is denoted gray.

Theorems & Definitions (54)

• Conjecture 1: dm-on-15
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9: dm-on-15