Topological graph theory
The graph genus problem is NP-complete
Journal of Algorithms
Approximation algorithms for NP-complete problems on planar graphs
Journal of the ACM (JACM)
A note on approximating graph genus
Information Processing Letters
Theoretical Computer Science - Special issue on computing and combinatorics
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Vertex cover: further observations and further improvements
Journal of Algorithms
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Which problems have strongly exponential complexity?
Journal of Computer and System Sciences
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The Dominating Set Problem Is Fixed Parameter Tractable for Graphs of Bounded Genus
SWAT '02 Proceedings of the 8th Scandinavian Workshop on Algorithm Theory
Planarization of Graphs Embedded on Surfaces
WG '95 Proceedings of the 21st International Workshop on Graph-Theoretic Concepts in Computer Science
On the existence of subexponential parameterized algorithms
Journal of Computer and System Sciences - Special issue on Parameterized computation and complexity
Parameterized Complexity
Subexponential parameterized algorithms on graphs of bounded-genus and H-minor-free graphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Polynomial-time data reduction for dominating set
Journal of the ACM (JACM)
Fast sub-exponential algorithms and compactness in planar graphs
ESA'11 Proceedings of the 19th European conference on Algorithms
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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).