Linear Algorithms for Partitioning Embedded Graphs of Bounded Genus

This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive $\varepsilon$ we show that any $n$-vertex graph $G$ of genus $g$ can be divided in $O(n+g)$ time into components whose sizes do not exceed $\varepsilon n$ by removing a set $C$ of $O(\sqrt{(g+1/\varepsilon)n})$ vertices. Our result improves the best previous ones with respect to the size of $C$ and the time complexity of the algorithm. Moreover, we show that one can cut off from $G$ a piece of no more than $(1-\varepsilon)n$ vertices by removing a set of $O(\sqrt{n\varepsilon (g\varepsilon +1})$) vertices. Both results are optimal up to a constant factor.

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