# On the Maximum Weight Independent Set Problem in graphs without induced cycles of length at least five

## TL;DR

This paper presents polynomial and subexponential algorithms for solving the Maximum Weight Independent Set problem in specific classes of graphs characterized by the absence of long holes and certain prisms, advancing understanding of graph algorithms.

## Contribution

It introduces the first polynomial-time algorithm for MWIS in long-hole-free graphs with no fixed k-prism and a subexponential algorithm for the general case.

## Key findings

- Polynomial-time algorithm for MWIS in long-hole-free graphs with no k-prism
- Subexponential algorithm for MWIS in long-hole-free graphs
- Application to maximum clique in certain perfect graphs

## Abstract

A hole in a graph is an induced cycle of length at least $4$, and an antihole is the complement of an induced cycle of length at least $4$. A hole or antihole is long if its length is at least $5$. For an integer $k$, the $k$-prism is the graph consisting of two cliques of size $k$ joined by a matching. The complexity of Maximum (Weight) Independent Set (MWIS) in long-hole-free graphs remains an important open problem. In this paper we give a polynomial time algorithm to solve MWIS in long-hole-free graphs with no $k$-prism (for any fixed integer $k$), and a subexponential algorithm for MWIS in long-hole-free graphs in general. As a special case this gives a polynomial time algorithm to find a maximum weight clique in perfect graphs with no long antihole, and no hole of length $6$. The algorithms use the framework of minimal chordal completions and potential maximal cliques.