Let $G=(V,E)$ be a connected graph. A subset $S\subset V$ is a cut of $G$ if $G-S$ is disconnected. A near triangulation is a 2-connected plane graph that has at most one face that is not a triangle. In this paper, we explore minimal cuts of 4-connected planar graphs. Our main result is that every minimal cut of a 4-connected planar graph $G$ is connected if and only if $G$ is a near-triangulation. We use this result to sketch a linear-time algorithm for finding a disconnected cut of a 4-connected planar graph.

翻译：设$G=(V,E)$为一个连通图。若$G-S$不连通，则子集$S\subset V$称为$G$的一个割集。近三角剖分是指至多有一个非三角形面的2-连通平面图。本文研究了4-连通平面图的极小割集。主要结论为：4-连通平面图$G$的每一个极小割集都是连通割集当且仅当$G$是近三角剖分图。利用该结论，我们勾勒出在4-连通平面图中寻找不连通割集的线性时间算法。