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 $Ω(kn)$ bits of space. They asked whether $Ω(kn)$ 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.

翻译：对于图$G=(V,E)$，$k$连通性预言机是一种数据结构，当给定$s,t \in V$时，它能判定$G$中是否存在至少$k+1$条内部不相交的$st$路径。对于无向图，Pettie、Saranurak和Yin [STOC 2022, pp. 151-161]证明了任何$k$连通性预言机都需要$Ω(kn)$比特的存储空间。他们提出疑问：若$G$是$k$连通的，是否仍需要$Ω(kn)$比特？我们将通过一个非常简洁的证明表明，即使$G$是$k$连通的，该下界依然成立，从而回答了这一开放性问题。