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

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