# Parameterized Algorithms for Maximum Cut with Connectivity Constraints

## TL;DR

This paper investigates two connectivity-constrained variants of the Maximum Cut problem, establishing their NP-completeness on planar graphs and providing fixed-parameter tractable algorithms based on graph parameters and solution size.

## Contribution

It proves NP-completeness of these variants on planar bipartite and split graphs and develops parameterized algorithms using clique-width, tree-width, twin-cover, and solution size.

## Key findings

- NP-complete on planar bipartite and split graphs
- Parameterized algorithms based on clique-width, tree-width, twin-cover
- FPT algorithms with respect to solution size

## Abstract

We study two variants of \textsc{Maximum Cut}, which we call \textsc{Connected Maximum Cut} and \textsc{Maximum Minimal Cut}, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity requirements. Both problems are known to be NP-complete even on planar graphs whereas \textsc{Maximum Cut} on planar graphs is solvable in polynomial time. We first show that these problems are NP-complete even on planar bipartite graphs and split graphs. Then we give parameterized algorithms using graph parameters such as clique-width, tree-width, and twin-cover number. Finally, we obtain FPT algorithms with respect to the solution size.