# Cluster Deletion on Interval Graphs and Split Related Graphs

## TL;DR

This paper proves that the Cluster Deletion problem can be solved in polynomial time on interval graphs, resolving a long-standing open problem, and explores its complexity on related graph classes.

## Contribution

It provides the first polynomial-time algorithm for Cluster Deletion on interval graphs and analyzes its complexity on subclasses of split graphs.

## Key findings

- Polynomial-time algorithm for Cluster Deletion on interval graphs.
- NP-completeness of Cluster Deletion on certain split graph generalizations.
- Two polynomial-time algorithms for subclasses of these generalizations.

## Abstract

In the {\sc Cluster Deletion} problem the goal is to remove the minimum number of edges of a given graph, such that every connected component of the resulting graph constitutes a clique. It is known that the decision version of {\sc Cluster Deletion} is NP-complete on ($P_5$-free) chordal graphs, whereas {\sc Cluster Deletion} is solved in polynomial time on split graphs. However, the existence of a polynomial-time algorithm of {\sc Cluster Deletion} on interval graphs, a proper subclass of chordal graphs, remained a well-known open problem. Our main contribution is that we settle this problem in the affirmative, by providing a polynomial-time algorithm for {\sc Cluster Deletion} on interval graphs. Moreover, despite the simple formulation of the algorithm on split graphs, we show that {\sc Cluster Deletion} remains NP-complete on a natural and slight generalization of split graphs that constitutes a proper subclass of $P_5$-free chordal graphs. To complement our results, we provide two polynomial-time algorithms for {\sc Cluster Deletion} on subclasses of such generalizations of split graphs.