Linkless and Flat Embeddings in 3-Space

Abstract

We consider piecewise linear embeddings of graphs in 3-space ℝ3. Such an embedding is linkless if every pair of disjoint cycles forms a trivial link (in the sense of knot theory). Robertson, Seymour and Thomas (J. Comb. Theory, Ser. B 64:185–227, 1995) showed that a graph has a linkless embedding in ℝ3 if and only if it does not contain as a minor any of seven graphs in Petersen’s family (graphs obtained from K 6 by a series of YΔ and ΔY operations). They also showed that a graph is linklessly embeddable in ℝ3 if and only if it admits a flat embedding into ℝ3, i.e. an embedding such that for every cycle C of G there exists a closed 2-disk D⊆ℝ3 with D∩G=∂D=C. Clearly, every flat embedding is linkless, but the converse is not true. We consider the following algorithmic problem associated with embeddings in ℝ3: Flat Embedding: For a given graph G, either detect one of Petersen’s family graphs as a minor in G, or return a flat (and hence linkless) embedding of G in ℝ3. The first outcome is a certificate that G has no linkless and no flat embeddings. Our main result is to give an O(n 2) algorithm for this problem. While there is a known polynomial-time algorithm for constructing linkless embeddings (van der Holst in J. Comb. Theory, Ser. B 99:512–530, 2009), this is the first polynomial-time algorithm for constructing flat embeddings in 3-space. This settles a problem proposed by Lovasz ( www.cs.elte.hu/~lovasz/klee.ppt, 2000).