Given a graph \(G=(V, E)\), a connected cut \(\delta (U)\) is the set of edges of E linking all vertices of U to all vertices of \(V\backslash U\) such that the induced subgraphs G[U] and \(G[V\backslash U]\) are connected. Given a positive weight function w defined on E, the connected maximum cut problem (CMAX CUT) is to find a connected cut \(\varOmega \) such that \(w(\varOmega )\) is maximum among all connected cuts. CMAX CUT is NP-hard even for planar graphs, and thus for graph without the excluded minor \(K_5\). In this paper, we prove that CMAX CUT is polynomial for the class of graphs without the excluded minor \(K_5\backslash e\), denoted by \({\mathcal {G}}(K_5\backslash e)\). We deduce two quadratic time algorithms: one for the minimum cut problem in \({\mathcal {G}}(K_5\backslash e)\) without computing the maximum flow, and another one for Hamilton cycle problem in the class of graphs without the two excluded minors the prism \(P_6\) and \(K_{3, 3}\). This latter problem is NP-complete for maximal planar graphs.

给定一个图\（G =（V，E）\），一个相连的切口\（\ delta（U）\）是E的一组边，将U的所有顶点链接到\（V \反斜杠U \ ），以使诱导子图G [ U ]和\（G [V \反斜杠U] \）相连。给定在E上定义的正权重函数w，连接的最大切割问题（CMAX CUT）是找到一个连接的切割\（\ varOmega \），使得\（w（\ varOmega）\）在所有连接的切割中最大。CMAX CUT即使对于平面图也是NP-hard，因此对于没有排除次要\（K_5 \）的图。在本文中，我们证明CMAX CUT是图类的多项式，没有排除的次要\（K_5 \反斜杠e \），用\（{\ mathcal {G}}（K_5 \反斜杠e）\）表示。我们推导了两种二次时间算法：一种用于\（{\ mathcal {G}}（K_5 \反斜杠e）\）中的最小割问题，而另一种用于在不具有图的情况下的汉密尔顿循环问题两个排除的未成年人棱镜\（P_6 \）和\（K_ {3，3} \）。对于最大平面图，后一个问题是NP完全的。