đ¤ AI Summary
This study addresses the problem of efficiently enumerating minimal removable vertex sets (MinRS) whose removal triggers core collapse in k-cores, thereby assessing network robustness under node deletions. Building upon a monotone system framework, the authors propose a general method to enumerate all MinRS for any k-core variant. The key innovation lies in the first formal definition of the "in-dominating seed property" and the proof that standard k-cores satisfy this property, which reduces the enumeration complexity from O((n+m)n) to O((n+m)log n). This yields the first O((n+m)log n)-delay algorithm for enumerating all k-core subgraphs. The approach naturally extends to weighted, multilayer, and (k,â)-core models, significantly outperforming the baseline algorithm by Boley et al. (2010).
đ Abstract
In network vulnerability analysis, it is crucial to evaluate the robustness of $k$-cores against vertex removals. A $k$-core is often fragile since removing a few vertices can trigger a large reduction in the core size, a phenomenon known as core collapse. In this paper, we study the problem of enumerating all minimal removable sets (MinRSs) of a given $k$-core, where a MinRS is a minimal nonempty set of vertices whose removal results in a smaller $k$-core graph. We consider this problem within a general mathematical framework based on monotone systems. We show that, for a monotone system that is given with an underlying graph $G=(V,E)$, all MinRSs of a solution can be enumerated in $O((n+m)nĪ_Ī)$ time, where $n=|V|$, $m=|E|$ and $Ī_Ī$ denotes the computation time of evaluating the monotone function of the system. Furthermore, if the system satisfies the newly defined in-dominating seed property, the complexity drops to $O((n+m) \log n \cdot Ī_Ī)$ time. We prove that standard $k$-cores in undirected graphs satisfy this property, enabling MinRS enumeration in $O((n+m)\log n)$ time, a significant improvement over the baseline. We also extend our framework to enumerate all solutions in a given monotone system. This yields an $O((n+m)\log n)$-delay algorithm for all $k$-core subgraphs, outperforming an algorithm given by [Boley et al., Theoretical Computer Science, 2010]. Our framework is applicable to various $k$-core extensions, including weighted $k$-cores, multi-layer $\boldsymbol{k}$-cores, and $(k,\ell)$-cores.