On $k$-connectivity oracles in $k$-connected graphs

Zeev Nutov

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arXiv · 2601.03643

On $k$-connectivity oracles in $k$-connected graphs

Zeev Nutov

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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.

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Zeev Nutov. 2026-01-07. On $k$-connectivity oracles in $k$-connected graphs. https://arxiv.org/abs/2601.03643

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