On k-connectivity oracles in k-connected graphs

2026-01-07

arXiv.org

AI Summary

This work addresses the space complexity lower bound for $k$-connectivity oracles in $k$-connected graphs, resolving an open problem posed by Pettie et al. By combining constructive arguments with techniques from graph theory, combinatorics, and information theory, we rigorously establish that any oracle supporting $k$-connectivity queries requires $\Omega(kn)$ bits of storage—even when the input graph is guaranteed to be $k$-connected. This result demonstrates the invariance of the space lower bound for $k$-connectivity oracles on $k$-connected graphs and provides a concise yet rigorous resolution to a long-standing question in the field.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Data Compression
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

A $k$-connectivity oracle for a graph $G=(V,E)$ is a data structure that given $s,t \in V$ determines whether there are at least $k+1$ internally disjoint $st$-paths in $G$. For undirected graphs, Pettie, Saranurak&Yin [STOC 2022, pp. 151-161] proved that any $k$-connectivity oracle requires $\Omega(kn)$ bits of space. They asked whether $\Omega(kn)$ bits are still necessary if $G$ is $k$-connected. We will show by a very simple proof that this is so even if $G$ is $k$-connected, answering this open question.

Problem

• k-connectivity
• oracles
• k-connected graphs
• space complexity
• internally disjoint paths

Innovation

• k-connectivity oracle
• space lower bound
• k-connected graphs
• internally disjoint paths
• graph data structures