Deleting Vertices to Graphs of Bounded Genus

We show that a problem of deleting a minimum number of vertices from a graph to obtain a graph embeddable on a surface of a given Euler genus is solvable in time $$2^{C_g \cdot k^2 \log k} n^{\mathcal {O}(1)}$$2Cg·k2logknO(1), where k is the size of the deletion set, $$C_g$$Cg is a constant depending on the Euler genus g of the target surface, and n is the size of the input graph. On the way to this result, we develop an algorithm solving the problem in question in time $$2^{\mathcal {O}((t+g) \log (t+g))} n$$2O((t+g)log(t+g))n given a tree decomposition of the input graph of width t. The results generalize previous algorithms for the surface being a sphere by Marx and Schlotter (Algorithmica 62(3–4):807–822, 2012. https://doi.org/10.1007/s00453-010-9484-z), Kawarabayashi (in: 50th annual IEEE symposium on foundations of computer science, FOCS 2009, IEEE Computer Society, pp 639–648, 2009. https://doi.org/10.1109/FOCS.2009.45) and Jansen et al. (in: Chekuri (ed) 25th annual ACM-SIAM symposium on discrete algorithms, SODA 2014, SIAM, pp 1802–1811, 2014. https://doi.org/10.1137/1.9781611973402.130).