Connected max cut is polynomial for graphs without the excluded minor $K_5\backslash e$

Abstract

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.