We present near-optimal algorithms for detecting small vertex cuts in the \({\textsf{CONGEST}}\) model of distributed computing. Despite extensive research in this area, our understanding of the vertex connectivity of a graph is still incomplete, especially in the distributed setting. To this date, all distributed algorithms for detecting cut vertices suffer from an inherent dependency in the maximum degree of the graph, \(\Delta \). Hence, in particular, there is no truly sub-linear time algorithm for this problem, not even for detecting a single cut vertex. We take a new algorithmic approach for vertex connectivity which allows us to bypass the existing \(\Delta \) barrier. As a warm-up to our approach, we show a simple \(\widetilde{O}(D)\)-round randomized algorithm for computing all cut vertices in a D-diameter n-vertex graph. This improves upon the \(O(D+\Delta /\log n)\)-round algorithm of [Pritchard and Thurimella, ICALP 2008]. Our key technical contribution is an \(\widetilde{O}(D)\)-round randomized algorithm for computing all cut pairs in the graph, improving upon the state-of-the-art \(O(\Delta \cdot D)^4\)-round algorithm by [Parter, DISC ’19]. Note that even for the considerably simpler setting of edge cuts, currently \(\widetilde{O}(D)\)-round algorithms are known only for detecting pairs of cut edges. Our approach is based on employing the well-known linear graph sketching technique [Ahn, Guha and McGregor, SODA 2012] along with the heavy-light tree decomposition of [Sleator and Tarjan, STOC 1981]. Combining this with a careful characterization of the survivable subgraphs, allows us to determine the connectivity of \(G {\setminus } \{x,y\}\) for every pair \(x,y \in V\), using \(\widetilde{O}(D)\)-rounds. We believe that the tools provided in this paper are useful for omitting the \(\Delta \)-dependency even for larger cut values.

我们提出了用于检测分布式计算模型\({\textsf{CONGEST}}\)中的小顶点切割的近乎最佳算法。尽管在这一领域进行了广泛的研究，但我们对图的顶点连通性的理解仍然不完整，特别是在分布式环境中。迄今为止，所有用于检测切割顶点的分布式算法都受到图最大度\(\Delta \)的固有依赖性的影响。因此，特别是，对于这个问题，没有真正的亚线性时间算法，甚至没有用于检测单切割顶点。我们采用一种新的顶点连接算法方法，这使我们能够绕过现有的\(\Delta \)障碍。作为我们方法的热身，我们展示了一个简单的\(\widetilde{O}(D)\)轮随机算法，用于计算D直径n顶点图中的所有切割顶点。这改进了[Pritchard 和 Thurimella，ICALP 2008] 的\(O(D+\Delta /\log n)\)轮算法。我们的关键技术贡献是一个\(\widetilde{O}(D)\)轮随机算法，用于计算图中的所有割对，改进了最先进的\ (O(\Delta \cdot D )^4\)轮算法由 [Parter, DISC '19] 提供。请注意，即使对于边缘切割的相当简单的设置，目前\(\widetilde{O}(D)\)已知的圆形算法仅用于检测成对的切割边缘。我们的方法基于采用众所周知的线性图绘制技术 [Ahn, Guha 和 McGregor, SODA 2012] 以及 [Sleator 和 Tarjan, STOC 1981] 的重轻树分解。将此与可生存子图的仔细表征相结合，使我们能够确定每对\(x,y \in V\)的 \(G {\setminus } \{x,y\}\)的连通性，使用\ (\widetilde{O}(D)\) - 轮。我们相信，本文提供的工具对于忽略\(\Delta \)依赖性非常有用，即使对于较大的切割值也是如此。