# Connected max cut is polynomial for graphs without $K_5\backslash e$ as a minor

## TL;DR

This paper proves that the connected maximum cut problem can be solved in polynomial time for graphs that do not contain a specific minor, namely $K_5$ minus an edge, and provides an efficient algorithm for it.

## Contribution

It establishes polynomial solvability of CMAX CUT for graphs excluding $K_5$ minus an edge as a minor, expanding understanding of graph classes where this problem is tractable.

## Key findings

- CMAX CUT is NP-hard in general but polynomial for certain graph classes.
- A quadratic time algorithm is developed for the minimum cut problem in these graphs.
- The results do not require computing maximum flow, simplifying the process.

## 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 $\Omega$ such that $w(\Omega)$ is maximum among all connected cuts. CMAX CUT is NP-hard even for planar graphs. In this paper, we prove that CMAX CUT is polynomial for graphs without $K_5\backslash e$ as a minor. We deduce a quadratic time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.