New Oracles and Labeling Schemes for Vertex Cut Queries

We study the succinct representations of vertex cuts by centralized oracles and labeling schemes. For an undirected n-vertex graph G = (V, E) and integer parameter f ≥ 1, the goal is supporting vertex cut queries: Given F ⊆ V with |F| ≤ f, determine if F is a vertex cut in G. In the centralized data structure setting, it is required to preprocess G into an f-vertex cut oracle that can answer such queries quickly, while occupying only small space. In the labeling setting, one should assign a short label to each vertex in G, so that a cut query F can be answered by merely inspecting the labels assigned to the vertices in F. While the “st cut variants” of the above problems have been extensively studied and are known to admit very efficient solutions, the basic (global) “cut query” setting is essentially open (particularly for f > 3). This work achieves the first significant progress on these problems. f-Vertex Cut Labels: Every n-vertex graph admits an f-vertex cut labeling scheme, where the labels have length of Õ(n^1−1/f) bits (when f is polylogarithmic in n). This nearly matches the recent lower bound given by Long, Pettie and Saranurak (SODA 2025). f-Vertex Cut Oracles: For f = O(log n), every n-vertex graph G admits f-vertex cut oracle with Õ(n) space and Õ(2^f) query time. We also show that our f-vertex cut oracles for every 1 ≤ f ≤ n are optimal up to n^o(1) factors (conditioned on plausible fine-grained complexity conjectures). If G is f-connected, i.e., when one is interested in minimum vertex cut queries, the query time improves to Õ(f²), for any 1 ≤ f ≤ n. Our f-vertex cut oracles are based on a special form of hierarchical expander decomposition that satisfies some “cut respecting” properties. Informally, we show that any n-vertex graph G can be decomposed into terminal vertex expander graphs that “capture” all cuts of size at most f in G. The total number of vertices in this graph collection is Õ(n). We are hopeful that this decomposition will have further applications (e.g., to the dynamic setting).