A Note on Approximating Graph Genus

Abstract This note, using simple combinatorial analysis, shows two interesting facts on the approximability of graph minimum genus embeddings: 1. (1) for any function ƒ(n) = O(n e ) , 0 ⩽ e G of n vertices into a surface of genus bounded by γ min (G) + ƒ(n) , unless P = NP ; and 2. (2) there is a linear time algorithm that embeds a graph G of n vertices into a surface of genus bounded by max {4 γ min ( G ), γ min ( G ) + 4 n }, where γ min ( G ) denotes the minimum genus of the graph G . An approximation algorithm with approximation ratio O(√ n ) for bounded degree graph embeddings is also presented.