Linkless and Flat Embeddings in 3-space in Quadratic Time

We consider embeddings of graphs in the 3-space R (all embeddings inthis paper are assumed to be piecewise linear). An embbeding of a graph in R3 is linkless if every pair of disjoint cycles forms a trivial link (in the senseof knot theory), i.e., each of the two cycles (in R) can be embedded in aclosed topological 2-disk disjoint from the other cycle. Robertson, Seymour andThomas [38] showed that a graph has a linkless embedding in R if and only ifit does not contain as a minor any of seven graphs in Petersen’s family (graphsobtained fromK6 by a series of YΔ and ΔY operations). They also showed thata graph is linklessly embeddable in R if and only if it admits a flat embeddingintoR, i.e. an embedding such that for every cycle C of G, there exists a closeddisk D ⊆ R 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 embeddingsof graphs in R: Flat and Linkless Embedding Input: A graph G.Output: Either detect one of Petersen’s family graphs as a minor in G or returna flat (and linkless) embedding in the 3-space. The first conclusion is a certificate that the given graph has no linkless andno flat embeddings. In this paper we give anO(n) algorithm for this problem.Our algorithm does not depend on minor testing algorithms.