Lower bounds on the obstacle number of graphs

Given a graph $G$, an {\em obstacle representation} of $G$ is a set of points in the plane representing the vertices of $G$, together with a set of connected obstacles such that two vertices of $G$ are joined by an edge if and only if the corresponding points can be connected by a segment which avoids all obstacles. The {\em obstacle number} of $G$ is the minimum number of obstacles in an obstacle representation of $G$. It is shown that there are graphs on $n$ vertices with obstacle number at least $\Omega({n}/{\log n})$.

💡 Research Summary

The paper investigates the obstacle number of a graph, a parameter introduced by Alpert, Koch and Laison (2010). An obstacle representation consists of a set of points in the plane (the vertices) together with a collection of polygonal obstacles; two vertices are adjacent exactly when the straight‑line segment joining them avoids all obstacles. The obstacle number obs(G) is the smallest possible number of obstacles that realize G.

The authors improve the quantitative understanding of this parameter. Earlier work only showed that graphs with arbitrarily large obstacle numbers exist, but gave no non‑trivial lower bound as a function of the number of vertices n. The main contributions are three theorems.

Theorem 1 gives an upper bound on the number of labelled n‑vertex graphs that can be represented with at most h obstacles, and more generally with obstacles whose total number of sides is s. The proof relies on two classical geometric facts:

1. Order‑type bound (Goodman–Pollack, 1986): the number of distinct order types of n points in general position is 2^{O(n log n)}. An order type encodes the orientation of every triple of points and therefore determines the combinatorial structure of any straight‑line drawing together with the placement of obstacle vertices.

2. Face‑complexity bound (Arkin et al., 1995): in a straight‑line drawing of an n‑vertex graph each face can be bounded by O(n log n) line‑segment sides, independent of the number of edges.

Using (2) the authors replace each obstacle by a slightly shrunken polygon that lies entirely inside a single face; consequently each obstacle has at most O(n log n) sides. Then the whole configuration (vertices plus all obstacle vertices) has at most N = n + s points, and by (1) there are at most 2^{O(N log N)} different order types. Hence the number of graphs admitting an h‑obstacle representation is at most 2^{O(h n log n)} (part (i)), and the number admitting obstacles with total side count s is at most 2^{O(n log n + s log s)} (part (ii)).

Theorem 2 extracts a lower bound from the counting argument. The total number of labelled graphs on n vertices is 2^{\binom{n}{2}}. If h = c·n/ log n for a sufficiently small constant c, then 2^{O(h n log n)} = 2^{O(c n²)} is strictly smaller than 2^{\binom{n}{2}} for large n. Therefore not all graphs can be represented with h obstacles, implying the existence of an n‑vertex graph with obstacle number at least Ω(n/ log n). The authors also give a probabilistic proof: a random graph G(n,½) almost surely contains many vertex‑disjoint k‑vertex induced subgraphs of obstacle number ≤ 1, and Lemma 6 shows that if obs(G) were too small this would happen too often, contradicting the bound from Theorem 1(i). Choosing k ≈ 5 log n yields the same Ω(n/ log n) lower bound with high probability.

Theorem 3 answers a question from the original obstacle‑number paper: for every natural h there exists a graph whose obstacle number is exactly h. Starting from the complete graph K_n (obstacle number 0) and deleting edges one by one, each deletion can increase the obstacle number by at most one (by inserting a tiny obstacle that blocks the removed edge). Hence as we delete edges until we reach a graph known to have obstacle number > h, we must pass through a graph with obstacle number precisely h. The same argument works for convex obstacles, segment obstacles, and other natural variants.

The paper also discusses special cases. For convex obstacles the same Ω(n/ log n) bound holds because each convex obstacle can be placed inside a face of complexity O(n log n). For segment obstacles (each obstacle is a single line segment) the total number of sides is s = 2n, so Theorem 1(ii) yields a much stronger bound: there exist graphs with segment‑obstacle number at least Ω(n²/ log n).

Finally, the authors note several open problems. The obstacle number of the complete graph K_n when obstacles are restricted to points is conjectured to be O(n), with the best known upper bound O(n² / √log n) due to Pach. Whether the obstacle number of all planar graphs is bounded by a constant remains unknown, although outerplanar graphs have been shown to have obstacle number at most 5. The general upper bound for an n‑vertex graph is trivially O(n²) (place a tiny obstacle for every non‑edge), and improving this bound is a major challenge.

In summary, the paper provides the first non‑trivial linear‑over‑logarithmic lower bound for the obstacle number of general graphs, establishes tight counting estimates for graphs with few obstacles, and settles the existence of graphs with any prescribed obstacle number. The techniques elegantly combine combinatorial geometry (order types, face complexity) with probabilistic method, opening new avenues for further research on visibility graphs and related geometric graph parameters.