Constructive results from graph minors: linkless embeddings

Abstract

A formal study of three-dimensional topological graph theory is initiated. The problem of deciding whether a graph can be embedded in 3-space so that no collection of vertex-disjoint cycles is topologically linked is considered first. The Robertson-Seymour Theory of Graph Minors is applicable to this problem and guarantees the existence of an O(n/sup 3/) algorithm for the decision problem. However, not even a finite-time decision procedure was known for this problem. A small set of forbidden minors for linkless embeddable graphs is exhibited, and it is shown that any graph with these minors cannot be embedded without linked cycles. It is further established that any graph that does not contain these minors is embeddable without linked cycles. Thus, an O(n/sup 3/) algorithm for the decision problem is demonstrated. It is believed that the proof technique will lead to an algorithm for actually embedding a graph, provided it does not contain the forbidden minors.<>