The Maximum Cut Problem in Co-bipartite Chain Graphs

TL;DR

This paper investigates the maximum cut problem in a specific subclass of co-bipartite graphs called co-bipartite chain graphs, providing explicit solutions and polynomial-time algorithms for certain cases.

Contribution

The paper introduces polynomial-time solutions for MaxCut in co-bipartite chain graphs, including the twin-free case, advancing understanding of MaxCut complexity in this class.

Findings

MaxCut is polynomial-time solvable in co-bipartite chain graphs.

Explicit solutions are provided for the twin-free case.

MaxCut remains NP-hard in general co-bipartite graphs.

Abstract

A \emph{co-bipartite chain} graph is a co-bipartite graph in which the neighborhoods of the vertices in each clique can be linearly ordered with respect to inclusion. It is known that the maximum cut problem (MaxCut) is NP-Hard in co-bipartite graphs. We consider MaxCut in co-bipartite chain graphs. We first consider the twin-free case and present an explicit solution. We then show that MaxCut is polynomial time solvable in this graph class.