# The independent set problem is FPT for even-hole-free graphs

## TL;DR

This paper proves that the maximum independent set problem is fixed-parameter tractable in even-hole-free graphs, advancing understanding of its computational complexity in this graph class.

## Contribution

It establishes the fixed-parameter tractability of MIS in even-hole-free graphs, a long-standing open problem, using a novel application of augmenting graph techniques.

## Key findings

- MIS is FPT in even-hole-free graphs.
- The approach uses augmenting graph techniques twice.
- Provides new insights into the complexity of MIS in special graph classes.

## Abstract

The class of even-hole-free graphs is very similar to the class of perfect graphs, and was indeed a cornerstone in the tools leading to the proof of the Strong Perfect Graph Theorem. However, the complexity of computing a maximum independent set (MIS) is a long-standing open question in even-hole-free graphs. From the hardness point of view, MIS is W[1]-hard in the class of graphs without induced 4-cycle (when parameterized by the solution size). Halfway of these, we show in this paper that MIS is FPT when parameterized by the solution size in the class of even-hole-free graphs. The main idea is to apply twice the well-known technique of augmenting graphs to extend some initial independent set.