An $f$-edge (or vertex) connectivity certificate is a sparse subgraph that maintains connectivity under the failure of at most $f$ edges (or vertices). It is well known that any $n$-vertex graph admits an $f$-edge (or vertex) connectivity certificate with $\Theta(f n)$ edges (Nagamochi and Ibaraki, Algorithmica 1992). A recent work by (Bodwin, Haeupler and Parter, SODA 2024) introduced a new and considerably stronger variant of connectivity certificates that can preserve connectivity under any failing set of edges with bounded degree. For every $n$-vertex graph $G=(V,E)$ and a degree threshold $f$, an $f$-Edge-Faulty-Degree (EFD) certificate is a subgraph $H \subseteq G$ with the following guarantee: For any subset $F \subseteq E$ with $deg(F)\leq f$ and every pair $u,v \in V$, $u$ and $v$ are connected in $H - F$ iff they are connected in $G - F$. For example, a $1$-EFD certificate preserves connectivity under the failing of any matching edge set $F$ (hence, possibly $|F|=\Theta(n)$). In their work, [BHP'24] presented an expander-based approach (e.g., using the tools of expander decomposition and expander routing) for computing $f$-EFD certificates with $O(f n \cdot poly(\log n))$ edges. They also provided a lower bound of $\Omega(f n\cdot \log_f n)$, hence $\Omega(n\log n)$ for $f=O(1)$. In this work, we settle the optimal existential size bounds for $f$-EFD certificates (up to constant factors), and also extend it to support vertex failures with bounded degrees (where each vertex is incident to at most $f$ faulty vertices). Specifically, we show that for every $n>f/2$, any $n$-vertex graph admits an $f$-EFD (and $f$-VFD) certificates with $O(f n \cdot \log(n/f))$ edges and that this bound is tight. Our upper bound arguments are considerably simpler compared to prior work, do not use expanders, and only exploit the basic structure of bounded degree edge and vertex cuts.

翻译：$f$边（或顶点）连通性证书是一种稀疏子图，它能在最多$f$条边（或顶点）发生故障时保持连通性。众所周知，任何$n$顶点图都存在一个具有$\Theta(f n)$条边的$f$边（或顶点）连通性证书（Nagamochi和Ibaraki，Algorithmica 1992）。(Bodwin, Haeupler和Parter, SODA 2024) 最近的工作引入了一种新的、更强大的连通性证书变体，它能在任何有界度的边故障集下保持连通性。对于每个$n$顶点图$G=(V,E)$和度阈值$f$，一个$f$边故障度（EFD）证书是一个子图$H \subseteq G$，并具有以下保证：对于任意满足$deg(F)\leq f$的边子集$F \subseteq E$以及每一对顶点$u,v \in V$，$u$和$v$在$H - F$中连通当且仅当它们在$G - F$中连通。例如，一个$1$-EFD证书可以在任何匹配边集$F$（因此可能$|F|=\Theta(n)$）发生故障时保持连通性。在他们的工作中，[BHP'24]提出了一种基于扩展图的方法（例如，使用扩展图分解和扩展图路由等工具）来计算具有$O(f n \cdot poly(\log n))$条边的$f$-EFD证书。他们还给出了$\Omega(f n\cdot \log_f n)$的下界，因此对于$f=O(1)$为$\Omega(n\log n)$。在本工作中，我们确定了$f$-EFD证书（在常数因子内）的最优存在性规模界限，并将其扩展到支持有界度的顶点故障（其中每个顶点最多与$f$个故障顶点相邻）。具体而言，我们证明对于所有$n>f/2$，任何$n$顶点图都存在具有$O(f n \cdot \log(n/f))$条边的$f$-EFD（以及$f$-VFD）证书，并且这个界限是紧的。与先前工作相比，我们的上界论证要简单得多，不使用扩展图，仅利用了有界度边割和顶点割的基本结构。