A topological space for which graph embeddability is undecidable (original) (raw)

1980, Journal of Combinatorial Theory, Series B

There exists a path-connected dability is undecidable. subspace of the plane for which graph embed-From Edmonds' Permutation Theorem ([ 3, 11) and a generalization due to Stahl [6], it follows that graph embeddability is decidable for all surfaces, orientable as well as nonorientable. We show the existence of a topological space G such that there is no algorithm to decide whether a finite graph is embeddable in G. In fact, G will be a path-connected subspace of the real plane. We begin by defining a set S of disjoint graphs G,, n > 3. The graph G, is obtained from the n-cycle C, by replacing each edge by two parallel edges. Clearly, each G, is 2-connected and planar. In addition, these graphs have a useful property which we prove as a lemma. LEMMA 1. No subdivision of G, is isomorphic to a subgraph of G, for m # n. Proof. This follows easily from the observation that, for each G,, every pair of adjacent vertices is connected by three disjoint paths, while nonadjacent vertices are connected by only two disjoint paths. 4