A connected graph is 4-connected if it contains at least five vertices and removing any three of them does not disconnect it. A frequent preprocessing step in graph drawing is to decompose a plane graph into its 4-connected components and to determine their nesting structure. A linear-time algorithm for this problem was already proposed by Kant. However, using common graph data structures, we found the subroutine dealing with triangulated graphs difficult to implement in such a way that it actually runs in linear time. As a drop-in replacement, we provide a different, easy-to-implement linear-time algorithm that decomposes a triangulated graph into its 4-connected components and computes the respective nesting structure. The algorithm is based on depth-first search.

翻译：一个连通图如果包含至少五个顶点，且删除其中任意三个顶点后仍保持连通，则称为4-连通图。在图绘制中，一个常见的预处理步骤是将平面图分解为其4-连通分量，并确定它们的嵌套结构。Kant已针对该问题提出了一种线性时间算法。然而，在使用通用图数据结构时，我们发现处理三角化图的子程序难以实现为真正线性时间运行。作为替代方案，我们提供了一种不同且易于实现的线性时间算法，该算法将三角化图分解为4-连通分量，并计算相应的嵌套结构。该算法基于深度优先搜索。