Connected max cut is polynomial for graphs without the excluded minor K5\e

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\U such that the induced subgraphs G[U] and G[V\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 Omega such that w(Omega) 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\e, denoted by G(K-5\e). We deduce two quadratic time algorithms: one for the minimum cut problem in G( K-5\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,K-3. This latter problem is NP-complete for maximal planar graphs.