A Tight Bound for Facial Distance Patterns in Planar Graphs

2026-08-07

AI Summary

This work addresses the problem of bounding the number of distinct distance-difference patterns induced by vertex sequences on a designated face in undirected, unweighted planar graphs. By combining combinatorial graph-theoretic analysis with distance-pattern modeling and leveraging AI-assisted proof techniques (via the GPT 5.6-Sol model), the authors improve the previously known $O(k^3)$ upper bound to a tight $O(k^2)$, thereby resolving a conjecture posed at ISAAC'22. This advancement yields several algorithmic consequences: it enhances the compression rate of Okamura–Seymour metric embeddings, leads to improved space complexity for constant-time exact distance oracles, enables more efficient distributed diameter computation, and—most notably—delivers the first centralized diameter algorithm running in $\widetilde{O}(n^{8/5})$ time, surpassing the prior $\widetilde{O}(n^{5/3})$ bound established for weighted directed graphs.

Abstract

Let $G$ be an undirected unweighted planar graph and let $S=(s_0,\dots,s_{k-1})$ be the vertices of a designated face, listed in cyclic order. Consider a vector that stores the distances from an arbitrary vertex $v$ to all vertices of $S$. The pattern of $v$ is obtained by taking the difference between every pair of consecutive values in this vector. Li and Parter [STOC'19] proved an upper bound of $O(k^3)$ on the number of unique patterns over all vertices of $G$. We improve this to $O(k^2)$, matching a known lower bound and settling a conjecture in [ISAAC'22]. The simple proof was found by OpenAI's GPT 5.6-Sol model. Plugging this new bound into known results has the following three immediate implications for undirected unweighted planar graphs: (1) it gives an improved compression of the Okamura-Seymour metric (2) it improves the space required by constant-time exact distance oracles, and (3) it improves the fastest distributed algorithm for computing the diameter. We further present a previously unknown and nontrivial implication: a (centralized) $\tilde{O}(n^{8/5})$-time algorithm for computing the diameter, improving over the $\tilde{O}(n^{5/3})$ algorithm of [SODA'18] which works for weighted directed planar graphs. Thus, there is currently a gap between the time for computing the diameter between weighted and unweighted planar graphs.

Problem

• planar graphs
• facial distance patterns
• distance vectors
• pattern complexity
• graph diameter

Innovation

• planar graphs
• distance patterns
• diameter algorithm
• tight bound
• distance oracles