# Deleting vertices to graphs of bounded genus

## TL;DR

This paper presents a fixed-parameter tractable algorithm for vertex deletion problems to obtain graphs embeddable on surfaces of bounded genus, generalizing previous sphere-based algorithms.

## Contribution

It introduces a new algorithm that efficiently solves the vertex deletion problem for graphs of bounded genus, extending prior work on sphere embeddings.

## Key findings

- Algorithm runs in time $2^{C_g \, k^2 \log k} n^{O(1)}$ for graphs of genus g.
- Develops an algorithm using tree decompositions with time $2^{O((t+g) \log (t+g))} n$.
- Generalizes previous algorithms for surface embeddings from sphere to higher genus surfaces.

## Abstract

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 $2^{C_g \cdot k^2 \log k} n^{O(1)}$, where $k$ is the size of the deletion set, $C_g$ 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 $2^{O((t+g) \log (t+g))} n$, 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 2012], Kawarabayashi [FOCS 2009], and Jansen, Lokshtanov, and Saurabh [SODA 2014].