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LINKLESS AND FLAT EMBEDDINGS IN 3-SPACE IN QUADRATIC TIME

By Ken-ichi Kawarabayashi, Stephan Kreutzer and Bojan Mohar

Abstract

We consider embeddings of graphs in the 3-space R 3 (all embeddings in this paper are assumed to be piecewise linear). An embbeding of a graph in R 3 is linkless if every pair of disjoint cycles forms a trivial link (in the sense of knot theory), i.e., each of the two cycles (in R 3) can be embedded in a closed topological 2-disk disjoint from the other cycle. Robertson, Seymour and Thomas [38] showed that a graph has a linkless embedding in R 3 if and only if it does not contain as a minor any of seven graphs in Petersen’s family (graphs obtained from K6 by a series of YΔ andΔYoperations). They also showed that a graph is linklessly embeddable in R 3 if and only if it admits a flat embedding into R 3, i.e. an embedding such that for every cycle C of G, there exists a closed disk D ⊆ R 3 with D ∩ G = ∂D = C. Clearly, every flat embeddings is linkless, but the converse is not true. We consider the following algorithmic problem associated with embedding

Topics: Linkless embedding, Petersen family, Flat embedding
Year: 2009
OAI identifier: oai:CiteSeerX.psu:10.1.1.417.6681
Provided by: CiteSeerX
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