We provide a deterministic algorithm for computing the $5$-edge-connected components of an undirected multigraph in linear time. There were probably good indications that this computation can be performed in linear time, but no such algorithm was actually known prior to this work. Thus, our paper answers a theoretical question, and sheds light on the possibility that a solution may exist for general $k$. A key component in our algorithm is an oracle for answering connectivity queries for pairs of vertices in the presence of at most four edge-failures. Specifically, the oracle has size $O(n)$, it can be constructed in linear time, and it answers connectivity queries in the presence of at most four edge-failures in worst-case constant time, where $n$ denotes the number of vertices of the graph. We note that this is a result of independent interest. Our paper can be considered as a follow-up of recent work on computing the $4$-edge-connected components in linear time. However, in dealing with the computation of the $5$-edge-connected components, we are faced with unique challenges that do not appear when dealing with lower connectivity. The problem is that the $4$-edge cuts in $3$-edge-connected graphs are entangled in various complicated ways, that make it difficult to organize them in a compact way. Here we provide a novel analysis of those cuts, that reveals the existence of various interesting structures. These can be exploited so that we can disentangle and collect only those cuts that are essential in computing the $5$-edge-connected components. This analysis may provide a clue for a general solution for the $k$-edge-connected components, or other related graph connectivity problems.

翻译：我们提出了一种确定性算法，可在线性时间内计算无向多重图的 $5$ 边连通分量。此前已有充分迹象表明该计算可在线性时间内完成，但实际算法在本工作之前尚未存在。因此，本文回应了一个理论问题，并揭示了存在针对一般 $k$ 的解决方案的可能性。算法的一个关键组件是一个用于在最多四条边失效情况下回答顶点对连通性查询的预言机。具体而言，该预言机大小为 $O(n)$，可在线性时间内构建，并在最坏情况下以常数时间回答最多四条边失效时的连通性查询（其中 $n$ 表示图的顶点数）。我们注意到这一结果具有独立意义。本文可视为近期关于线性时间计算 $4$ 边连通分量工作的后续研究。但在处理 $5$ 边连通分量计算时，我们面临了低连通性情形中未出现的独特挑战。问题在于 $3$ 边连通图中的 $4$ 边割以多种复杂方式纠缠，难以对其进行紧凑组织。本文提供了这些割的新颖分析，揭示了多种有趣结构的存在。利用这些结构，我们能够解缠并仅收集对计算 $5$ 边连通分量至关重要的割。这一分析可能为一般 $k$ 边连通分量或其他相关图连通性问题提供解决方案的线索。