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Embedding a Graph Into the Torus in Linear Time
, 1994
"... A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph\Omega of G of small branch size that cannot be embedded in the torus. 1 Introduction Let K be a subgraph of G, and suppose that we are ..."
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A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph\Omega of G of small branch size that cannot be embedded in the torus. 1 Introduction Let K be a subgraph of G, and suppose that we are given an embedding of K in some surface. The embedding extension problem asks whether it is embedding extension problem possible to extend the embedding of K to an embedding of G in the same surface, and any such embedding is an embedding extension of K to G. An embedding extension obstruction for embedding extensions is a subgraph\Omega of G \Gamma E(K) such that obstruction the embedding of K cannot be extended to K [ \Omega\Gamma The obstruction is small small if K [\Omega is homeomorphic to a graph with a small number of edges. If\Omega is small, then it is easy to verify (for example, by checking all the possibilities Supported in part by the Ministry of Science and Technolo...
A Simpler and Faster Torus Embedding Algorithm
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A Faster Algorithm for Torus Embedding
, 2006
"... Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and di ..."
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Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and discuss how it was inspired by the quadratic time planar embedding algorithm of Demoucron, Malgrange and Pertuiset. We show that it is faster in practice than the only fully implemented alternative (also exponential) and explain how both the algorithm itself and the knowledge gained during its development might be used to solve the wellstudied problem of finding the complete set of torus obstructions.
An Efficient Algorithm for the Genus with Explicit Construction of Forbidden Problem Subgraphs
"... We give an algorithm for irnbedding a graph G of n vertices onto an oriented surface of minimum genus g. If g z O then we also construct a forbidden subgraph of G which is homecjmorphic to a graph of size exp(O(g)! ) which cannot be irnbedded on a surface of genus g1. Our algorithm takes sequential ..."
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We give an algorithm for irnbedding a graph G of n vertices onto an oriented surface of minimum genus g. If g z O then we also construct a forbidden subgraph of G which is homecjmorphic to a graph of size exp(O(g)! ) which cannot be irnbedded on a surface of genus g1. Our algorithm takes sequential time exp(O(g)!)nO(l). Since exp(O (g)!) = exp(exp(O (glob))), our algorithm is polynomial time for genus g = O(loglog(n)/logloglog(,m)). A simple parallel implementation of our algorithm takes parallel time (logn)O(l J+O(g)! using exp(O(g)!)nO(l) processors. We give also the smallest known upper bound, namely exp(O(g)!), on the number F(g) of homomorphic distinct forbidden subgraphs for graph imbeddings onto a surface of genus g. Previous algorithms for imbedding a graph onto a surface of genus g had the following sequential time bounds: n ‘(n) for the naive algorithm, n 0 (g) for the algorithm of [Fil!otti, Miller, Reif,79], and f(g)n2 by the algorithm of [Robertson and Seymour,86), where f(g) is a function only of g. The celebrated work of [Robertson and Seymour,86] also gave the first known finite bound for F(g). However their proof spanned many papers and was highly nonconstructive; f(g) and F(g) were bounded by some (large) tower of exponents of g. Our work provides a distinct constructive appxoach giving considerably improved bounds for f(g) and F(g) and vastly simplified proofs. In particular, we use a “bootstrap ” tdmique that uses a discovered forbidden subgraph for given genus g’cg to aid us in determination of genus g ‘+1 imbeddings. It seems likely that our techniques can be extended to many other problems on graphs with bounded tree width.