  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 $|S|=V$.
