Visual Abstract

A (Γ1, Γ2)-labeled graph is an oriented graph with its edges labeled by elements of the direct sum of two groups Γ1, Γ2. A cycle in such a labeled graph is (Γ1, Γ2)-non-zero if it is non-zero in both coordinates. Our main result is a generalization of the Flat Wall Theorem of Robertson and Seymour to (Γ1, Γ2)-labeled graphs. As an application, we determine all canonical obstructions to the Erdős-Pósa property for (Γ1, Γ2)-non-zero cycles in (Γ1, Γ2)-labeled graphs. The obstructions imply that the halfintegral Erdős-Pósa property always holds for (Γ1, Γ2)-non-zero cycles.Moreover, our approach gives a unified framework for proving packing results for constrained cycles in graphs. For example, as immediate corollaries we recover the Erdős-Pósa property for cycles and S-cycles and the half-integral Erdős-Pósa property for odd cycles and odd S-cycles. Furthermore, we recover Reed's Escher-wall Theorem.We also prove many new packing results as immediate corollaries. For example, we show that the half-integral Erdős-Pósa property holds for cycles not homologous to zero, odd cycles not homologous to zero, and Scycles not homologous to zero. Moreover, the (full) Erdős-Pósa property holds for S1-S2-cycles and cycles not homologous to zero on an orientable surface. Finally, we also describe the canonical obstructions to the Erdős-Pósa property for cycles not homologous to zero and for odd S-cycles.

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 Erdős-Pósa property in graphs embedded in a fixed surface. This extends the fact that odd cycles satisfy the Erdős-Pósa 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.

Let W t denote the wheel on t + 1 vertices. We prove that for every integer t ≥ 3 there is a constant c = c(t) such that for every integer k ≥ 1 and every graph G, either G has k vertex-disjoint subgraphs each containing W t as a minor, or there is a subset X of at most ck log k vertices such that G − X has no W t minor. This is best possible, up to the value of c. We conjecture that the result remains true more generally if we replace W t with any fixed planar graph H.

Abstract. We give an O(g 1/2 n 3/2 +g 3/2 n 1/2 )-size extended formulation for the spanning tree polytope of an n-vertex graph embedded in a surface of genus g, improving on the known O(n 2 + gn)-size extended formulations following from Wong [10] and Martin [7].