This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive " 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 "n by removing a set C of O( p (g + 1=")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 o from G a piece of no more than (1 ?")n vertices by removing a set of O( p n"(g" + 1) vertices. Both results are optimal up to a constant factor.
More generally, for graphs embedded on genus g surfaces, a separator with O(~/~) vertices can be found in O(n) time REF .
Linear Algorithms for Partitioning Embedded Graphs of Bounded Genus
SIAM Journal of Discrete Mathematics
Mathematics Computer Science