Shortest Path Separators in Unit Disk Graphs

TL;DR

This work resolves an open problem by showing that every unit-disk graph admits a shortest path $1$-neighborhood separator: removing the $1$-neighborhood of two shortest paths plus the $1$-neighborhood of two additional vertices yields a balanced cut. The authors develop a novel framework based on non-crossing path systems formed from Delaunay triangulation edges and extend Lipton–Tarjan’s planar separator ideas to triangulated perturbations of these path systems. Key technical components include the definition and non-crossing properties of Delaunay paths, a Path Extension Lemma to preserve non-crossing extensions, and the crominating property of Delaunay edges. The result improves previous $3$-neighborhood separator bounds, is optimal in the sense that a $0$-neighborhood separator can fail for cliques, and provides a foundation for further divide-and-conquer approaches in unit-disk graphs with practical implications for geometric network design. The separator can be constructed in $O(n^2)$ time, with the path system itself of size $O(n^2)$, and the perturbation-based LT separator lies at the core of the approach.

Abstract

We introduce a new balanced separator theorem for unit-disk graphs involving two shortest paths combined with the 1-hop neighbours of those paths and two other vertices. This answers an open problem of Yan, Xiang and Dragan [CGTA '12] and improves their result that requires removing the 3-hop neighborhood of two shortest paths. Our proof uses very different ideas, including Delaunay triangulations and a generalization of the celebrated balanced separator theorem of Lipton and Tarjan [J. Appl. Math. '79] to systems of non-intersecting paths.

Figures (5)

• Figure 1: (Left) The points $u, v, x, y\in S$ are drawn with circles of radius $1/2$. The unique shortest path tree in $G$ with starting vertex $u$ has a crossing edge. (Right) Two reflected copies results in a graph where no plane shortest path tree exists, regardless of starting vertex.
• Figure 2: An example of a forward crossing between the red and blue chain. The cloud obscures the shared middle part of the polygonal chains
• Figure 4: The paths $\Pi[w_1]$ (in blue) and $\Pi[w_2]$ (in red) for $w_1, w_2\in W_{a-1}$ don't cross. However, the Delaunay path $\Delta[u]$ consisting of $u\in W_a$ to $w_1$ (in green) concatenated with $\Pi[w_1]$ forms a path that crosses $\Pi[w_2]$.
• Figure 6: (Left) A path system $\Pi$ to a vertex $s$ in a Delaunay triangulation. (Middle) A perturbation of all points on all paths at vertices other than $s$ by at most $\varepsilon$ such that no two paths intersect except at $s$. (Right) A triangulation of the perturbation of the paths using only previously existing edges.
• Figure 7: Suppose an edge $xy\in E$ intersects but is not dominated by a Delaunay edge $uv$. Without loss of generality assume that disk $x$ intersects edge $uv$ and lies above $uv$. We show that $x\in D^\uparrow_{uv}$ and $y\in D^\downarrow_x\subset D^\downarrow_{uv}$.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2: Balanced shortest path separator of Lipton-Tarjan LiptonT1979
• Lemma 3
• Lemma 4: Dickerson-Drysdale DickersonD90; Cabello-Jejčič CabelloJ15
• Lemma 5
• Lemma 6
• Lemma 7: Path Extension Lemma
• Lemma 8
• Lemma 9: Spanning non-crossing path systems of psuedo-shortest paths
• Lemma 10