Obstacles, Slopes, and Tic-Tac-Toe: An excursion in discrete geometry and combinatorial game theory

A drawing of a graph is said to be a {\em straight-line drawing} if the vertices of $G$ are represented by distinct points in the plane and every edge is represented by a straight-line segment connecting the corresponding pair of vertices and not passing through any other vertex of $G$. The minimum number of slopes in a straight-line drawing of $G$ is called the slope number of $G$. We show that every cubic graph can be drawn in the plane with straight-line edges using only the four basic slopes ${0,\pi/4,\pi/2,-\pi/4}$. We also prove that four slopes have this property if and only if we can draw $K_4$ with them. 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 obstacles (connected polygons) 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 $G$. We show that there are graphs on $n$ vertices with obstacle number $\Omega({n}/{\log n})$. We show that there is an $m=2n+o(n)$, such that, in the Maker-Breaker game played on $\Z^d$ where Maker needs to put at least $m$ of his marks consecutively in one of $n$ given winning directions, Breaker can force a draw using a pairing strategy. This improves the result of Kruczek and Sundberg who showed that such a pairing strategy exits if $m\ge 3n$. A simple argument shows that $m$ has to be at least $2n+1$ if Breaker is only allowed to use a pairing strategy, thus the main term of our bound is optimal.

💡 Research Summary

The dissertation “Obstacles, Slopes, and Tic‑Tac‑Toe: An excursion in discrete geometry and combinatorial game theory” presents three independent but thematically linked results concerning the representation of graphs under severe resource constraints and the analysis of a positional game on the integer lattice.

1. Slope number of cubic graphs
The slope number sl(G) of a graph G is the smallest number of distinct slopes needed in a straight‑line drawing of G. While it is known that graphs of maximum degree ≥5 can have arbitrarily large slope numbers, the exact value for 3‑regular (cubic) graphs remained open. The author proves that every connected cubic graph admits a straight‑line drawing using only the four basic slopes {0, π/4, π/2, −π/4}. The proof proceeds in several stages. First, a stronger statement is shown for sub‑cubic graphs (maximum degree ≤3 with at least one vertex of degree ≤2): by fixing the x‑coordinates of low‑degree vertices to a linearly independent set of real numbers and carefully ordering vertices, one can embed the graph using only the four basic directions while respecting a set of north‑south placement constraints. Next, any cubic graph containing triangles is reduced to a triangle‑free cubic graph via a local replacement lemma. For triangle‑free cubic graphs the previous embedding works, yielding a drawing with the four basic slopes. The paper also establishes an equivalence: a set S of four slopes is “good” (i.e., works for all cubic graphs) if and only if S is an affine image of the basic four slopes, which in turn holds exactly when K₄ can be drawn with S. Consequently, four slopes are both necessary (K₄ requires four) and sufficient for all cubic graphs.

2. Lower bound on obstacle number
Given a set of points representing vertices and a collection of polygonal obstacles, the visibility graph consists of all pairs of points whose connecting segment avoids every obstacle. The obstacle number obs(G) is the minimum number of obstacles needed to realize G as a visibility graph. Prior work gave only a modest Ω(√log n) lower bound. Using a probabilistic construction based on random graphs G(n, ½), the author shows that for infinitely many n there exist graphs with obs(G) ≥ c·n/ log n for some constant c>0. The argument observes that each obstacle can block only O(log n) edges on average, so to eliminate the O(n²) non‑edges of a dense random graph one needs at least Ω(n/ log n) obstacles. The result is later strengthened to Ω(n² log n) when obstacles are restricted to line segments, demonstrating that the obstacle number can be substantially larger than previously thought.

3. Maker–Breaker game on ℤᵈ with pairing strategy
The paper studies a Maker–Breaker positional game played on the d‑dimensional integer lattice. A set of n winning directions is fixed; Maker wins by occupying m consecutive points in any one direction. Breaker aims to prevent this. Kruczek and Sundberg proved that if m ≥ 3n then Breaker can force a draw using a pairing strategy, and they conjectured that m ≥ 2n+1 should already suffice. The author confirms this conjecture asymptotically: for any fixed n, there exists m = 2n + o(n) (more precisely, m−1 = p where p is a prime ≥ 2n+1) such that Breaker has a pairing strategy guaranteeing a draw. The construction partitions ℤᵈ into blocks of size p and pairs each lattice point with a partner inside the same block. Because p is prime, the arithmetic progression of any winning direction cycles through all residues modulo p, ensuring that each potential winning line contains at least one paired pair, which Breaker can block. This yields a bound that matches the known necessary condition up to an additive constant, showing that the pairing strategy is essentially optimal.

Overall significance
The work unifies three strands of discrete mathematics: geometric graph drawing (slope number), visibility representations (obstacle number), and positional game theory (pairing strategies). Each result pushes the known limits—establishing exact slope requirements for cubic graphs, providing a substantially stronger obstacle‑number lower bound, and nearly optimal conditions for Breaker’s pairing strategy. The methods blend combinatorial constructions, probabilistic arguments, and elementary number theory, offering tools that are likely to influence further research in graph drawing, geometric representations, and combinatorial games.