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 g ₁(n) is fixed parameter tractable if and only if g ₁(n) = o(n ²), and Dominating Set on graphs of genus bounded by g ₂(n) is fixed parameter tractable if and only if g ₂(n) = n ^o(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 g ₃(n) are solvable in subexponential time if and only if g ₃(n) = o(n).