Let $\kappa(s,t)$ denote the maximum number of internally disjoint $st$-paths in an undirected graph $G$. We consider designing a compact data structure that answers $k$-bounded node connectivity queries: given $s,t \in V$ return $\min\{\kappa(s,t),k+1\}$. A trivial data structure has space $O(n^2)$ and query time $O(1)$. A data structure of Hsu and Lu has space $O(k^2n)$ and query time $O(\log k)$,and a randomized data structure of Iszak and Nutov has space $O(kn\log n)$ and query time $O(k \log n)$. We extend the Hsu-Lu data structure to answer queries in time $O(1)$. In parallel to our work, Pettie, Saranurak and Yin extended the Iszak-Nutov data structure to answer queries in time $O(\log n)$. Our data structure is more compact for $k<\log n$, and our query time is always better. We then augment our data structure by a list of cuts that enables to return a pointer to a minimum $st$-cut in the list (or to a cut of size $\leq k$) whenever $\kappa(s,t) \leq k$. A trivial data structure has cut list size $n(n-1)/2$, and cut query time $O(1)$, while the Pettie, Saranurak and Yin data structure has list size $O(kn \log n)$ and cut query time $O(\log n)$. We show that $O(kn)$ cuts suffice to return an $st$-cut of size $\leq k$, and a list of $O(k^2 n)$ cuts contains a minimum $st$-cut for every $s,t \in V$. In the case when $S$ is a node subset with $\kappa(s,t) \geq k$ for all $s,t \in V$, we show that $3|S|$ cuts suffice, and that these cuts can be partitioned into $O(k)$ laminar families. Thus using space $O(kn)$ we can answers each connectivity and cut queries for $s,t \in S$ in $O(1)$ time, generalizing and substantially simplifying the proof of a result of Pettie and Yin for the case $|S|=V$.

翻译：设$\kappa(s,t)$表示无向图$G$中内部不相交$st$-路径的最大数目。我们考虑设计一种紧凑的数据结构，用于回答$k$-有界节点连通性查询：给定$s,t \in V$，返回$\min\{\kappa(s,t),k+1\}$。一种平凡的数据结构具有空间复杂度$O(n^2)$和查询时间复杂度$O(1)$。Hsu和Lu的数据结构具有空间复杂度$O(k^2n)$和查询时间复杂度$O(\log k)$，而Iszak和Nutov的随机化数据结构具有空间复杂度$O(kn\log n)$和查询时间复杂度$O(k \log n)$。我们将Hsu-Lu数据结构扩展为$O(1)$时间复杂度的查询。与我们工作并行的是，Pettie、Saranurak和Yin将Iszak-Nutov数据结构扩展为$O(\log n)$时间复杂度的查询。当$k<\log n$时，我们的数据结构更紧凑，且查询时间复杂度始终更优。随后，我们通过一个割列表对该数据结构进行增强，使得当$\kappa(s,t) \leq k$时，能够返回指向列表中最小$st$-割（或大小$\leq k$的割）的指针。一种平凡的数据结构具有$n(n-1)/2$大小的割列表和$O(1)$的割查询时间复杂度，而Pettie、Saranurak和Yin的数据结构具有$O(kn \log n)$大小的列表和$O(\log n)$的割查询时间复杂度。我们证明$O(kn)$个割足以返回大小$\leq k$的$st$-割，且包含$O(k^2 n)$个割的列表可覆盖所有$s,t \in V$的最小$st$-割。当$S$是节点子集且对所有$s,t \in V$均有$\kappa(s,t) \geq k$时，我们证明$3|S|$个割即可满足要求，且这些割可划分为$O(k)$个层状族。因此，使用$O(kn)$空间即可在$O(1)$时间内回答$S$中任意$s,t$的连通性和割查询，这推广并显著简化了Pettie和Yin关于$|S|=V$情形结论的证明。