# Subexponential Parameterized Algorithms for Cut and Cycle Hitting Problems on H-Minor-Free Graphs

We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on H-minor free graphs. In particular, we obtain the following results (where k is the solution-size parameter). 1. 2^O(√(k)log k)· n^O(1) time algorithms for Edge Bipartization and Odd Cycle Transversal; 2. a 2^O(√(k)log^4 k)· n^O(1) time algorithm for Edge Multiway Cut and a 2^O(r √(k)log k)· n^O(1) time algorithm for Vertex Multiway Cut, where r is the number of terminals to be separated; 3. a 2^O((r+√(k))log^4 (rk))· n^O(1) time algorithm for Edge Multicut and a 2^O((√(rk)+r) log (rk))· n^O(1) time algorithm for Vertex Multicut, where r is the number of terminal pairs to be separated; 4. a 2^O(√(k)log g log^4 k)· n^O(1) time algorithm for Group Feedback Edge Set and a 2^O(g √(k)log(gk))· n^O(1) time algorithm for Group Feedback Vertex Set, where g is the size of the group. 5. In addition, our approach also gives n^O(√(k)) time algorithms for all above problems with the exception of n^O(r+√(k)) time for Edge/Vertex Multicut and (ng)^O(√(k)) time for Group Feedback Edge/Vertex Set. We obtain our results by giving a new decomposition theorem on graphs of bounded genus, or more generally, an h-almost-embeddable graph for any fixed constant h. In particular we show the following. Let G be an h-almost-embeddable graph for a constant h. Then for every p∈ℕ, there exist disjoint sets Z_1,…,Z_p ⊆ V(G) such that for every i ∈{1,…,p} and every Z'⊆ Z_i, the treewidth of G/(Z_i\ Z') is O(p+|Z'|). Here G/(Z_i\ Z') is the graph obtained from G by contracting edges with both endpoints in Z_i \ Z'.

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