Genus characterizes the complexity of certain graph problems: Some tight results

We study the fixed-parameter tractability, subexponential time computability, and approximability of the well-known NP-hard problems: independent set, vertex cover, and dominating set. We derive tight results and show that the computational complexity of these problems, with respect to the above complexity measures, is dependent on the genus of the underlying graph. For instance, we show that, under the widely-believed complexity assumption W[1]≠ FPT, independent set on graphs of genus bounded by g1(n) is fixed parameter tractable if and only if g1(n)=o(n2), and dominating set on graphs of genus bounded by g2(n) is fixed parameter tractable if and only if g2(n)=no(1). Under the assumption that not all SNP problems are solvable in subexponential time, we show that the above three problems on graphs of genus bounded by g3(n) are solvable in subexponential time if and only if g3(n)=o(n). We also show that the independent set, the kernelized vertex cover, and the kernelized dominating set problems on graphs of genus bounded by g4(n) have PTAS if g4(n)=o(n/logn), and that, under the assumption P ≠ NP, the independent set problem on graphs of genus bounded by g5(n) has no PTAS if g5(n)=Ω(n), and the vertex cover and dominating set problems on graphs of genus bounded by g6(n) have no PTAS if g6(n)=nΩ(1).