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APPEARED IN BULLETIN OF THE

arXiv:math/9301216v1 [math.CO] 1 Jan 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 1, January 1993, Pages 84-89LINKLESS EMBEDDINGS OF GRAPHS IN 3-SPACENeil Robertson, P. D. Seymour, and Robin ThomasAbstract. We announce results about flat (linkless) embeddings of graphs in 3-space.A piecewise-linear embedding of a graph in 3-space is called flat if everycircuit of the graph bounds a disk disjoint from the rest of the graph.We haveshown:(i) An embedding is flat if and only if the fundamental group of the complementin 3-space of the embedding of every subgraph is free.

(ii) If two flat embeddings of the same graph are not ambient isotopic, then theydiffer on a subdivision of K5 or K3,3. (iii) Any flat embedding of a graph can be transformed to any other flat embeddingof the same graph by “3-switches”, an analog of 2-switches from the theory of planarembeddings.

In particular, any two flat embeddings of a 4-connected graph are eitherambient isotopic, or one is ambient isotopic to a mirror image of the other. (iv) A graph has a flat embedding if and only if it has no minor isomorphic toone of seven specified graphs.

These are the graphs that can be obtained from K6by means of Y ∆- and ∆Y -exchanges.1. IntroductionAll spatial embeddings are assumed to be piecewise linear.

If C, C′ are disjointsimple closed curves in S3, then their linking number, lk(C, C′), is the number oftimes (mod 2) that C crosses over C′ in a regular projection of C ∪C′. In this papergraphs are finite, undirected, and may have loops and multiple edges.

Every graphis regarded as a topological space in the obvious way. We say that an embeddingof a graph G in S3 is linkless if every two disjoint circuits of G have zero linkingnumber.

The following is a result of Sachs [13, 14] and Conway and Gordon [3]. (1.1)The graph K6 (the complete graph on six vertices) has no linkless embedding.Proof.

Let φ be an embedding of K6 into S3. By studying the effect of a crossingchange in a regular projection, it is easy to see that the mod 2 sum P lk(φ(C1), φ(C2)),where the sum is taken over all unordered pairs of disjoint circuits C1, C2 of K6, isan invariant independent of the embedding.

By checking an arbitrary embeddingwe can establish that this invariant equals 1.□Let G be a graph and let v be a vertex of G of valency 3 with distinct neighbors.Let H be obtained from G by deleting v and adding an edge between every pair1991 Mathematics Subject Classification. Primary 05C10, 05C75, 57M05, 57M15, 57M25.Received by the editors January 14, 1992 and, in revised form, May 12, 1992The first author’s research was performed under a consulting agreement with Bellcore; he wassupported by NSF under Grant No.

DMS-8903132 and by ONR under Grant No. N00014-911-J-1905.

The third author was supported in part by NSF under Grant No. DMS-9103480, and inpart by DIMACS Center, Rutgers University, New Brunswick, New Jersey 08903c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2N. Robertson, P. D. Seymour, and R. Thomasof neighbors of v. We say that H is obtained from G by a Y ∆-exchange and thatG is obtained from H by a ∆Y -exchange.

The Petersen family is the set of allgraphs that can be obtained from K6 by means of Y ∆- and ∆Y -exchanges. Thereare exactly seven such graphs, one of which is the Petersen graph.

Pictures of thesegraphs can be found in [13–15]. Sachs [13, 14] has in fact shown that no member ofthe Petersen family has a linkless embedding [the argument is similar to the proofof (1.1)] and raised the problem of characterizing linklessly embeddable graphs.

Agraph is a minor of another if the first can be obtained from a subgraph of thesecond by contracting edges. It is easy to see that the property of having a linklessembedding is preserved under taking minors, and that led Sachs to conjecture thata graph is linklessly embeddable if and only if it has no minor in the Petersenfamily.

We have shown that this is true. Moreover, let us say that an embeddingφ of a graph G in S3 is flat if for every circuit C of G there exists an open diskin S3 disjoint from φ(G) whose boundary is φ(C).

Clearly every flat embeddingis linkless, but the converse need not hold. However, B¨ohme [1] and Saran [15]conjectured that a graph has a linkless embedding if and only if it has a flat one.This is also true, for we have shown the following.

(1.2)For a graph G, the following are equivalent:(i) G has a flat embedding,(ii) G has a linkless embedding,(iii) G has no minor in the Petersen family.There have been a number of other attempts [8, 15, 2] at proving (iii) ⇒(i) and(iii) ⇒(ii). However, none of them is correct.For the proof of (1.2) we need the following two theorems, which may be ofindependent interest.

(1.3)Let φ be an embedding of a graph G in S3. Then φ is flat if and only if forevery subgraph G′ of G, the fundamental group of S3 −φ(G′) is free.Let φ1, φ2 be two embeddings of a graph G in S3.

We say that φ1, φ2 are ambientisotopic if there exists an orientation preserving homeomorphism h of S3 onto S3such that φ1 = hφ2. (We remark that by a result of Fisher [4] h can be realized byan ambient isotopy.) If φ is an embedding of a graph G in S3 we denote by −φ theembedding of G obtained by composing φ with the antipodal map.

(1.4)Let G be a 4-connected graph and let φ1, φ2 be two flat embeddings of G.Then φ1 is ambient isotopic to either φ2 or −φ2.2. The fundamental groupA basic tool for working with flat embeddings is the following lemma of B¨ohme[1] (see also [15]).

(2.1)Let φ be a flat embedding of a graph G into S3, and let C1, C2, . .

. , Cn be afamily of circuits of G such that for every i ̸= j, the intersection of Ci and Cj is ei-ther connected or null.

Then there exist pairwise disjoint open disks D1, D2, . .

. , Dn,disjoint from φ(G) and such that φ(Ci) is the boundary of Di for i = 1, 2, .

. .

, n.We illustrate the use of (2.1) with the following, which is a special case of atheorem of Wu [18]. An embedding φ of a graph G in S3 is spherical if there existsa surface Σ ⊆S3 homeomorphic to S2 such that φ(G) ⊆Σ.

Clearly if φ is sphericalthen G is planar.

LINKLESS EMBEDDINGS OF GRAPHS IN 3-SPACE3(2.2)Let φ be an embedding of a planar graph G in S3. Then φ is flat if and onlyif it is spherical.Proof.

Clearly if φ is spherical then it is flat. We prove the converse only for thecase when G is 3-connected.

Let C1, C2, . .

. , Cn be the collection of face-boundariesin some planar embedding of G. These circuits satisfy the hypothesis of (2.1).

LetD1, D2, . .

. , Dn be the disks as in (2.1); then φ(G)∪D1 ∪D2 ∪· · ·∪Dn is the desiredsphere.□The following is a result of Scharlemann and Thompson [16].

(2.3)Let φ be an embedding of a graph G in S3. Then φ is spherical if and onlyif(i) G is planar, and(ii) for every subgraph G′ of G, the fundamental group of S3 −φ(G′) is free.We see that by (2.2), (1.3) is a generalization of (2.3).

In fact, we prove (1.3) byreducing it to planar graphs and then applying (2.3). Let us prove the “only if” partof (1.3).

Let G′ be a subgraph of G such that π1(S3 −φ(G′)) is not free. Choose amaximal forest F of G′ and let G′′ be obtained from G′ by contracting all edges of F,and let φ′′ be the induced embedding of G′′.

Then π1(S3−φ′′(G′′)) = π1(S3−φ(G′))is not free, but G′′ is planar, and so φ′′ is not flat by (2.2) and (2.3). Hence φ isnot flat, as desired.Let G be a graph, and let e be an edge of G. We denote by G\e(G/e) the graphobtained from G by deleting (contracting) e. If φ is an embedding of G in S3, thenit induces embeddings of G\e and (up to ambient isotopy) of G/e in the obviousway.

We denote these embeddings by φ\e and φ/e, respectively. (2.4)Let φ be an embedding of a graph G into S3, and let e be a nonloop edge ofG.

If both φ\e and φ/e are flat, then φ is flat.Proof. Suppose that φ is not flat.

By (1.3) there exists a subgraph G′ of G such thatπ1(S3 −φ(G′)) is not free. If e ̸∈E(G′) then φ\e is not flat by (1.3).

If e ∈E(G′)then φ/e is not flat by (1.3), because π1(S3 −(φ/e)(G′/e)) = π1(S3 −φ(G′)) is notfree.□We say that a graph G is a coforest if every edge of G is a loop. The followingfollows immediately from (2.4).

(2.5)Let φ be an embedding of a graph G in S3. Then φ is flat if and only if theinduced embedding of every coforest minor of G is flat.3.

UniquenessA graph H is a subdivision of a graph G if H can be obtained from G by replacingedges by pairwise internally-disjoint paths. We recall that Kuratowski’s theorem[6] states that a graph is planar if and only if it contains no subgraph isomorphicto a subdivision of K5 or K3,3.

It follows from a theorem of Mason [7] and (2.2)that any two flat embeddings of a planar graph are ambient isotopic. On the otherhand we have the following.

4N. ROBERTSON, P. D. SEYMOUR, AND R. THOMAS(3.1)The graphs K5 and K3,3 have exactly two nonambient isotopic flat embed-dings.Sketch of proof.

Let G be K3,3 or K5, let e be an edge of G, and let H be G\e.Notice that H is planar. From (2.1) it follows that if φ is a flat embedding ofG, then there is an embedded 2-sphere Σ ⊆S3 with φ(G) ∩Σ = φ(H).If φ1and φ2 are flat embeddings of G, we may assume (by replacing φ2 by an ambientisotopic embedding) that this 2-sphere Σ is the same for both φ1 and φ2.

Now φ1 isambient isotopic to φ2 if and only if φ1(e) and φ2(e) belong to the same componentof S3 −Σ.□As a curiosity we deduce that a graph has a unique flat embedding if and onlyif it is planar.We need the following three lemmas. We denote by f|X the restriction of amapping f to a set X.

(3.2)Let φ1, φ2 be two flat embeddings of a graph G that are not ambient isotopic.Then there exists a subgraph H of G isomorphic to a subdivision of K5 or K3,3 forwhich φ1|H and φ2|H are not ambient isotopic.We denote the vertex-set and edge-set of a graph G by V (G) and E(G) re-spectively.Let G be a graph and let H1, H2 be subgraphs of G isomorphic tosubdivisions of K5 or K3,3. We say that H1 and H2 are 1-adjacent if there existi ∈{1, 2} and a path P in G such that P has only its endvertices in common withHi and such that H3−i is a subgraph of the graph obtained from Hi by adding P.We say that H1 and H2 are 2-adjacent if there are seven vertices u1, u2, .

. .

, u7 ofG and thirteen paths Lij of G (1 ≤i ≤4 and 5 ≤j ≤7, or i = 3 and j = 4), suchthat(i) each path Lij has ends ui, uj,(ii) the paths Lij are mutually vertex-disjoint except for their ends,(iii) H1 is the union of Lij for i = 2, 3, 4 and j = 5, 6, 7, and(iv) H2 is the union of Lij for i = 1, 3, 4 and j = 5, 6, 7. (Notice that if H1 and H2 are 2-adjacent then they are both isomorphic to subdi-visions of K3,3 and that L34 is used in neither H1 nor H2.) We denote by K(G) thesimple graph with vertex-set all subgraphs of G isomorphic to subdivisions of K5or K3,3 in which two distinct vertices are adjacent if they are either 1-adjacent or2-adjacent.

The following is easy to see, using (3.1). (3.3)Let φ1, φ2 be two flat embeddings of a graph G, and let H, H′ be two adjacentvertices of K(G).If φ1|H is ambient isotopic to φ2|H, then φ1|H′ is ambientisotopic to φ2|H′.The third lemma is purely graph-theoretic.

(3.4)If G is a 4-connected graph, then K(G) is connected.We prove (3.4) in [10] by proving a stronger result, a necessary and sufficientcondition for H, H′ ∈V (K(G)) to belong to the same component of K(G) in anarbitrary graph G. The advantage of this approach is that it permits an inductiveproof using the techniques of deleting and contracting edges.Proof of (1.4). If G is planar then φ1 is ambient isotopic to φ2 by Mason’s theorem.Otherwise there exists, by Kuratowski’s theorem, a subgraph H of G isomorphic to

LINKLESS EMBEDDINGS OF GRAPHS IN 3-SPACE5a subdivision of K5 or K3,3. By replacing φ2 by −φ2 we may assume by (3.1) thatφ1|H is ambient isotopic to φ2|H.

From (3.3) and (3.4) we deduce that φ1|H′ isambient isotopic to φ2|H′ for every H′ ∈V (K(G)). By (3.2) φ1 and φ2 are ambientisotopic, as desired.□We now state a generalization of (1.4).

Let φ be a flat embedding of a graphG, and let Σ ⊆S3 be a surface homeomorphic to S2 meeting φ(G) in a set Acontaining at most three points. In one of the open balls into which Σ divides S3,say B, choose an open disk D with boundary a simple closed curve ∂D such thatA ⊆∂D ⊆Σ.

Let φ′ be an embedding obtained from φ by taking a reflection ofφ through D in B and leaving φ unchanged in Σ −B. We say that φ′ is obtainedfrom φ by a 3-switch.

The following analog of a theorem of Whitney [17] generalizes(1.4). (3.5)Let φ1, φ2 be two flat embeddings of a graph G in S3.Then φ2 can beobtained from φ1 by a series of 3-switches.4.

Main theoremThe difficult part of (1.2) is to show that (iii) implies (i).Let us just verybriefly sketch the main idea of the proof.Suppose that G is a minor-minimalgraph with no flat embedding. We first show that a Y ∆-exchange preserves theproperty of having a flat embedding; thus we may assume that G has no triangles(and indeed has some further properties that we shall not specify here).

It can beshown that G satisfies a certain weaker form of 5-connectivity. Suppose that thereare two edges e, f of G so that G\e/f and G/e/f are “Kuratowski 4-connected”.

(Kuratowski 4-connectivity is a slight weakening of 4-connectivity for which (1.4)still remains true.) Since G is minor-minimal with no flat embedding, there are flatembeddings φ1, φ2, φ3 of G\e, G/e, G/f, respectively.

Since G\e/f and G/e/f areboth Kuratowski 4-connected, we can assume (by replacing φ1 or φ2 or both byits mirror image) that φ1/f is ambient isotopic to φ3\e and that φ2/f is ambientisotopic to φ3/e. Now it can be argued (the details are quite complicated, see [12])that the uncontraction of f in φ1/f ≃φ3\e is the same as in φ2/f ≃φ3/e.

Let φbe obtained from φ3 by doing this uncontraction; then φ\e is ambient isotopic toφ1 and φ/e is ambient isotopic to φ2. Since both these embeddings are flat, φ isflat by (2.4), a contradiction.

Thus no two such edges e, f exist. But now a purelygraph-theoretic argument [11] (using the nonexistence of such edges e, f, the highconnectivity of G, and that the graph obtained from G by deleting v is nonplanarfor every vertex v of G) implies G has a minor in the Petersen family.Finally we would like to mention some algorithmic aspects of flat embeddings.

In[16] Scharlemann and Thompson describe an algorithm to test if a given embeddingis spherical. Using their algorithm, (2.2), and (2.5), we can test if a given embeddingis flat, by testing the flatness of all coforest minors.

At the moment there is noknown polynomial-time algorithm to test if an embedding of a given coforest is flat,because it includes testing if a knot is trivial. On the other hand, we can test intime O(|V (G)|3) if a given graph G has a flat embedding.

This is done by testingthe absence of minors isomorphic to members of the Petersen family, using thealgorithm [9] of the first two authors.References1. T. B¨ohme, On spatial representations of graphs, Contemporary Methods in Graph Theory

6N. ROBERTSON, P. D. SEYMOUR, AND R. THOMAS(R. Bodendieck, ed.

), Mannheim, Wien, Zurich, 1990, pp. 151–167.2., Lecture at the AMS Summer Research Conference on Graph Minors, Seattle, WA,June 1991.3.

J. H. Conway and C. McA. Gordon, Knots and links in spatial graphs, J. Graph Theory7 (1983), 445–453.4.

G. M. Fisher, On the group of all homeomorphisms of a manifold, Trans. Amer.

Math.Soc. 97 (1960), 193–212.5.

D. W. Hall, A note on primitive skew curves, Bull. Amer.

Math. Soc.

49 (1943), 935–937.6. C. Kuratowski, Sur le probl`eme des courbes gauches en topologie, Fund.

Math. 15 (1930),271–283.7.

W. K. Mason, Homeomorphic continuous curves in 2-space are isotopic in 3-space, Trans.Amer. Math.

Soc. 142 (1969), 269–290.8.

R. Motwani, A. Raghunathan, and H. Saran, Constructive results from graph minors:Linkless embeddings, Proc. 29th Symposium on the Foundations of Computer Science,Yorktown Heights, 1988.9.

N. Robertson and P. D. Seymour, Graph minors. XIII.

The disjoint paths problem, sub-mitted.10. N. Robertson, P. D. Seymour, and R. Thomas, Kuratowski chains, submitted.11., Petersen family minors, submitted.12., Sachs’ linkless embedding conjecture, manuscript.13.

H. Sachs, On spatial representation of finite graphs (Proceedings of a conference held in Lag´ow, February 10–13, 1981, Poland), Lecture Notes in Math., vol. 1018, Springer-Verlag,Berlin, Heidelberg, New York, and Tokyo, 1983.14., On spatial representations of finite graphs, finite and infinite sets, (A. Hajnal,L.

Lov´asz, and V. T. S´os, eds), Colloq. Math.

Soc. J´anos Bolyai, vol.

37, North-Holland,Budapest, 1984, pp. 649–662.15.

H. Saran, Constructive results in graph minors: Linkless embeddings, Ph.D. thesis, Uni-versity of California at Berkeley, 1989.16. M. Scharlemann and A. Thompson, Detecting unknotted graphs in 3-space, J. DifferentialGeom.

34 (1991), 539–560.17. H. Whitney, 2-isomorphic graphs, Amer.

J. Math.

55 (1933), 245–254.18. Y.-Q.

Wu, On planarity of graphs in 3-manifolds, Comment. Math.

Helv. (to appear).Department of Mathematics, Ohio State University, 231 West 18th Avenue, Colum-bus, Ohio 43210E-mail address: robertso@function.mps.ohio-state.eduBellcore, 445 South Street, Morristown, New Jersey 07962E-mail address: pds@bellcore.comSchool of Mathematics, Georgia Institute of Technology, Atlanta, Georgia30332E-mail address: thomas@math.gatech.edu

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