We study connectivity functions, that is, integer-valued symmetric submodular functions on a finite ground set attaining $0$ on the empty set. For a connectivity function $f$ on an $n$-element set $V$ and an integer $k\ge 0$, we show that the family of all sets $X\subseteq V$ with $f(X)=k$ admits a polynomial-size representation: it can be described by a list of at most $O(n^{4k})$ items, each consisting of a set to be included, another set to be excluded, and a partition of remaining elements, such that the union of some members of the partition and the set to be included are precisely all sets $X$ with $f(X)=k$. We also give an algorithm that constructs this representation in time $O(n^{2k+7}γ+n^{2k+8}+n^{4k+2})$, where $γ$ is the oracle time to evaluate $f$. This generalizes the low rank structure theorem of Bojańczyk, Pilipczuk, Przybyszewski, Sokołowski, and Stamoulis [Low rank MSO, arXiv, 2025] on cut-rank functions on graphs to general connectivity functions. As an application, for fixed $k$, we obtain a polynomial-time algorithm for finding a set $A$ with $f(A)=k$ and a prescribed cardinality constraint on $A$.

翻译：我们研究连通性函数，即定义在有限基集上、在空集上取值为0的整数值对称子模函数。对于定义在n元集合V上的连通性函数f和整数k≥0，我们证明所有满足f(X)=k的集合X⊆V的族具有多项式大小的表示：它可由至多O(n^{4k})个项组成的列表描述，每个项包含一个需包含的集合、一个需排除的集合以及剩余元素的一个划分，使得该划分中某些成员的并集与需包含的集合恰好构成所有满足f(X)=k的集合X。我们还给出一个算法，在O(n^{2k+7}γ+n^{2k+8}+n^{4k+2})时间内构造该表示，其中γ是评估f的预言机时间。这推广了Bojańczyk、Pilipczuk、Przybyszewski、Sokołowski和Stamoulis [Low rank MSO, arXiv, 2025] 关于图上割秩函数的低秩结构定理至一般连通性函数。作为应用，对于固定k，我们获得一个多项式时间算法，用于在A上给定基数约束的条件下寻找满足f(A)=k的集合A。