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 2Cg·k 2 log knO(1), where k is the size of the deletion set, 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 2O((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 [10], Kawarabayashi [6], and Jansen, Lokshtanov, and Saurabh [5].