# The stable set problem in graphs with bounded genus and bounded odd cycle packing number

## TL;DR

This paper presents a polynomial-time algorithm for the stable set problem in graphs with bounded genus and odd cycle packing number, extending known properties and providing efficient solutions for a long-standing complexity question.

## Contribution

It introduces a polynomial-time algorithm for stable set problems in graphs embedded in bounded genus surfaces and extends Erdős-Pósa property to such graphs.

## Key findings

- Polynomial-time algorithm for stable set in bounded genus graphs.
- Extended Erdős-Pósa property for 2-sided odd cycles in these graphs.
- Polynomial-size extended formulations for stable set polytopes.

## Abstract

Consider the family of graphs without $ k $ node-disjoint odd cycles, where $ k $ is a constant. Determining the complexity of the stable set problem for such graphs $ G $ is a long-standing problem. We give a polynomial-time algorithm for the case that $ G $ can be further embedded in a (possibly non-orientable) surface of bounded genus. Moreover, we obtain polynomial-size extended formulations for the respective stable set polytopes. To this end, we show that $2$-sided odd cycles satisfy the Erd\H{o}s-P\'osa property in graphs embedded in a fixed surface. This extends the fact that odd cycles satisfy the Erd\H{o}s-P\'osa property in graphs embedded in a fixed orientable surface (Kawarabayashi & Nakamoto, 2007). Eventually, our findings allow us to reduce the original problem to the problem of finding a minimum-cost non-negative integer circulation of a certain homology class, which turns out to be efficiently solvable in our case.