Connectivity Certificate against Bounded-Degree Faults: Simpler, Better and Supporting Vertex Faults

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 Θ(fn) 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 ⊆ G with the following guarantee: For any subset F ⊆ E with deg(F) ≤ f and every pair u, v ∈ 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| = Θ(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(fn·poly(log n)) edges. They also provided a lower bound of Ω(fn · log_f n), hence Ω(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(fn·log(n/f)) edges. 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.