Journal of the ACM (JACM)
An efficient algorithm for determining whether a cubic graph is toroidal
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Graphs and Hypergraphs
Graph Theory With Applications
Graph Theory With Applications
The page number of genus g graphs is (g)
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Finding a maximum-genus graph imbedding
Journal of the ACM (JACM)
Nonconstructive tools for proving polynomial-time decidability
Journal of the ACM (JACM)
SPAA '89 Proceedings of the first annual ACM symposium on Parallel algorithms and architectures
An efficient algorithm for the genus problem with explicit construction of forbidden subgraphs
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Upward planar drawing of single source acyclic digraphs
SODA '91 Proceedings of the second annual ACM-SIAM symposium on Discrete algorithms
The pagenumber of genus g graphs is O(g)
Journal of the ACM (JACM)
A shortest path approach to wireframe to solid model conversion
SMA '95 Proceedings of the third ACM symposium on Solid modeling and applications
Embedding graphs in an arbitrary surface in linear time
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Computing crossing numbers in quadratic time
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Creating volume models from edge-vertex graphs
SIGGRAPH '82 Proceedings of the 9th annual conference on Computer graphics and interactive techniques
Isomorphism of graphs with bounded eigenvalue multiplicity
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Isomorphism for graphs embeddable on the projective plane
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Computing crossing numbers in quadratic time
Journal of Computer and System Sciences - STOC 2001
Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Computing crossing number in linear time
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Maximum s-t-flow with k crossings in O(k3n log n) time
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Graph and map isomorphism and all polyhedral embeddings in linear time
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Errors in graph embedding algorithms
Journal of Computer and System Sciences
Compact systems for T-join and perfect matching polyhedra of graphs with bounded genus
Operations Research Letters
Weakly bipartite graphs and the Max-cut problem
Operations Research Letters
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In this paper we present an algorithm which on input a graph G and a positive integer g finds an embedding of G on a surface on genius g, if such an embedding exists. This algorithm runs in (v) O(g) steps where v is the number of vertices of G. We believe that removing the nondiscrete topological definitions (i.e., the notation or differentiability, 2-dimensional surface, etc.) from our formal definitions has a multitude of advantages. First our goal is to produce an algorithm which operates on discrete machines and thus at some point we must remove these notions anyway. Secondly, demonstrations on proofs in the amalgam of graph theory and topology have been riddled with flaws (e.g., 4-color theorem, planarity algorithms, Jordan curve theorem), and which, no doubt, this paper also suffers. The hope is that a combinatorial proof may transcend these problems. Third, our main goal is not just to draw graphs on “inner tubes” but to understand how graph theory, topology and computational complexity interact. We have kept no definitions sacred and we have redefined the notion of a graph. We have even rewritten Euler's formula.