We study a new and stronger notion of fault-tolerant graph structures whose size bounds depend on the degree of the failing edge set, rather than the total number of faults. For a subset of faulty edges $F \subseteq G$, the faulty-degree $\deg(F)$ is the largest number of faults in $F$ incident to any given vertex. We design new fault-tolerant structures with size comparable to previous constructions, but which tolerate every fault set of small faulty-degree $\deg(F)$, rather than only fault sets of small size $|F|$. Our main results are: - New FT-Certificates: For every $n$-vertex graph $G$ and degree threshold $f$, one can compute a connectivity certificate $H \subseteq G$ with $|E(H)| = \widetilde{O}(fn)$ edges that has the following guarantee: for any edge set $F$ with faulty-degree $\deg(F)\leq f$ and every vertex pair $u,v$, it holds that $u$ and $v$ are connected in $H \setminus F$ iff they are connected in $G \setminus F$. This bound on $|E(H)|$ is nearly tight. Since our certificates handle some fault sets of size up to $|F|=O(fn)$, prior work did not imply any nontrivial upper bound for this problem, even when $f=1$. - New FT-Spanners: We show that every $n$-vertex graph $G$ admits a $(2k-1)$-spanner $H$ with $|E(H)| = O_k(f^{1-1/k} n^{1+1/k})$ edges, which tolerates any fault set $F$ of faulty-degree at most $f$. This bound on $|E(H)|$ optimal up to its hidden dependence on $k$, and it is close to the bound of $O_k(|F|^{1/2} n^{1+1/k} + |F|n)$ that is known for the case where the total number of faults is $|F|$ [Bodwin, Dinitz, Robelle SODA '22]. Our proof of this theorem is non-constructive, but by following a proof strategy of Dinitz and Robelle [PODC '20], we show that the runtime can be made polynomial by paying an additional $\text{polylog } n$ factor in spanner size.

翻译：我们研究了一种新型且更强的容错图结构，其规模边界取决于故障边集的度数，而非故障总数。对于故障边子集 $F \subseteq G$，故障度 $\deg(F)$ 定义为与任一顶点关联的 $F$ 中故障边的最大数量。我们设计了新型容错结构，其规模与先前构造相当，但能容忍每个小故障度 $\deg(F)$ 的故障集，而不仅仅是小规模 $|F|$ 的故障集。主要成果如下： - 新型容错证书：对于任意 $n$ 顶点图 $G$ 和度数阈值 $f$，可计算连通性证书 $H \subseteq G$，边数为 $|E(H)| = \widetilde{O}(fn)$，具有以下保证：对于任意故障度 $\deg(F)\leq f$ 的边集 $F$ 及任意顶点对 $u,v$，$u$ 和 $v$ 在 $H \setminus F$ 中连通当且仅当它们在 $G \setminus F$ 中连通。该 $|E(H)|$ 边界几乎紧确。由于我们的证书能处理规模高达 $|F|=O(fn)$ 的故障集，先前工作即使对 $f=1$ 的情况也未给出该问题的任何非平凡上界。 - 新型容错稀疏图结构：我们证明任意 $n$ 顶点图 $G$ 均存在 $(2k-1)$-稀疏图 $H$，边数为 $|E(H)| = O_k(f^{1-1/k} n^{1+1/k})$，能容忍任意故障度至多 $f$ 的故障集 $F$。该 $|E(H)|$ 边界在 $k$ 的隐藏依赖关系下最优，且接近已知故障总数为 $|F|$ 情况下的边界 $O_k(|F|^{1/2} n^{1+1/k} + |F|n)$ [Bodwin, Dinitz, Robelle SODA '22]。该定理的证明是非构造性的，但遵循 Dinitz 和 Robelle [PODC '20] 的证明策略，我们表明若在稀疏图规模上增加 $\text{polylog } n$ 因子，则运行时可为多项式时间。