?WHAT IS... a Graph Minor Bojan Mohar A minor of a graph G describes a substructure of conjecture that every Kn -minor-free graph is G that is more general than a subgraph. If we take (n − 1)-colorable. Hadwiger’s Conjecture is still a subgraph of G and then contract some connected wide open and is one of the most challenging prob- pieces in this subgraph to single points, the re- lems in graph theory. sulting graph is called a minor of G. Formally, a At the beginning of the 1980s, Neil Robertson graph M is a minor of another graph G if G con- and Paul Seymour developed the theory of graph tains pairwise disjoint trees Tv ⊆ G, one for each minors in a series of twenty long papers. It took vertex v of M, such that whenever u and v are ad- twenty-one years [2, 3] to publish this seminal jacent vertices of M, there is an edge in G joining work, which had a tremendous impact not only on Tu and Tv. Closely related to this definition are the various branches of graph theory but also on many operations of deletion and contraction of an edge, other areas, most notably theoretical computer each of which clearly produces a minor. Conversely, science. The ultimate result of Robertson and Sey- we can obtain any minor M of a graph G, by first mour was the proof of Wagner’s Conjecture [3] deleting all edges except those in subgraphs Tv that (finite) graphs are well-quasi-ordered by the (v ∈ V(M) ) and those connecting T , T for u v graph minor relation, which is equivalent to the fol- uv ∈ E(M), and secondly contracting the edges in lowing statement: the trees Tv. In every infinite collection of graphs, there are two such that one is a minor of the other. A class M of graphs is minor-closed if for every G ∈M, all minors of G are also in M. Examples of minor-closed classes are the collection of all K K 3,3 5 planar graphs and, more generally, all graphs that Figure 1. The Kuratowski graphs K3,3 and K5. can be embedded in a fixed surface. Every minor- closed class M can be described by specifying the The first important appearance of graph mi- set of all minor-minimal graphs that are not in M nors is in the following version of the Kuratowski — these graphs are called the forbidden minors for Theorem: M. The well-quasi-ordering of graphs with respect A graph G can be embedded in the plane to the minor relation is equivalent to the follow- (is planar) if and only if neither the com- ing important result: plete graph K5 nor the complete bipar- tite graph K3,3 is a minor of G. For every minor-closed family of graphs, the set of forbidden minors is finite. In topology, this theorem is usually expressed in an equivalent form saying that no subgraph of G This is a far-reaching generalization of the Ku- is homeomorphic to K5 or K3,3. ratowski Theorem with a slight disadvantage that Wagner proved in 1937 that the Four Color The- its proof is nonconstructive—it does not yield a orem is equivalent to the statement that the ver- procedure for finding the forbidden minors for tices of every graph without K5 minors can be col- the family M, nor does it yield a constructive ored with four colors so that adjacent vertices bound on the number of forbidden minors. receive different colors. This led Hadwiger to Robertson and Seymour also proved that for every fixed graph M, there is an algorithm that has Bojan Mohar is professor of mathematics at the Univer- 3 sity of Ljubljana and a Canada Research Chair holder at time complexity O(n ) and that for a given graph Simon Fraser University, Burnaby, Canada. His email ad- G of order n decides if M is a minor of G. This yields dress is [email protected]. another powerful result: 338 NOTICES OF THE AMS VOLUME 53, NUMBER 3 For every minor-closed family M of genus (so that M itself cannot be embedded in S), graphs there exists a cubic time algo- and a bounded number of other “path-like” graphs rithm for testing membership in M— that are attached to G only along the face bound- one simply checks if a given graph con- aries on S. This shows that up to “small” and “well- tains some forbidden minor for M. behaved” perturbations, graphs with minor M ex- cluded are essentially 2-dimensional. This is a For instance, any of the following questions can surprising fact—graph minors have been intro- be solved in O(n3) time, where n is the order of the duced with the aim of proving and generalizing the input graph G: Kuratowski theorem, and yet, the structure theory (1) Let Σ be a fixed surface and G a graph. Can G of graph minors cannot be done without studying be embedded in Σ? graphs embedded in surfaces. (2) Is G linklessly embeddable in R3 The theory of graph minors has several power- (3) Is G knotlessly embeddable in R3 ? ful consequences for theoretical computer science, A linkless embedding of a graph is one where most notably in computational complexity and in no two cycles are linked, and a knotless embedding the theory of algorithms. Besides the aforemen- is one where no cycle is knotted. It is known that tioned consequences of well-quasi-ordering, the the set of forbidden minors for linkless embeddings concept of “tree-like” structure, formally giving consists of precisely seven graphs which can be ob- rise to the notion of the tree-width of graphs, has tained from the Petersen graph by a sequence of been widely applied in the theory of algorithms. Y∆ and ∆Y operations (the “Petersen family”). On Extensive research is being carried out today to the other hand, only sporadic examples of forbid- extend graph minor results to matroids. The goal den minors for knotless embeddings are known in this area is to prove a conjecture of Gian-Carlo (one of them is K ). 7 Rota, claiming that matroids representable over The set of forbidden minors is not known for any finite field are well-quasi-ordered with respect embeddings of graphs in general surfaces. Excep- to matroid minors. Part of this extensive project tions are the sphere, whose forbidden minors are has already been accomplished by Jim Geelen, Bert K5 and K3,3 by the Kuratowski Theorem, and the Gerards, and Geoff Whittle. projective plane, which has 35 forbidden minors. In closing, let us briefly return to edge-deletions Computer searches have shown that there are sev- and edge-contractions since they are closely re- eral thousands of forbidden minors for embed- lated to some other aspects of graph minors. In the dings in the torus, and there is no indication how 1950s, Tutte introduced a two-variable polyno- to obtain the complete list. Despite these difficul- mial, now known as the Tutte polynomial, defined ties, in some specific cases, like testing embedda- for a graph G recursively as follows. If G has no bility of graphs in a fixed surface, efficient algo- edges, then T (G; x, y)=1. Otherwise, let e be an rithms have been devised using different edge of G . If e is a loop, then T (G; x, y)= approaches. yT(G − e; x, y) ; if e is a cutedge, then T (G; x, y)= Another significant consequence of Robertson xT(G/e; x, y); otherwise T (G; x, y)=T (G − e; x, y)+ and Seymour’s work concerns the related structure. T (G/e; x, y) . It can be shown that the Tutte poly- Wagner proved that graphs that do not contain K5 nomial is well-defined for arbitrary graphs. Its co- as a minor are precisely those that can be built from efficients are nonnegative integers, and it turns planar graphs and subgraphs of a certain cubic out that T (G; x, y) is essentially the same as the par- graph on 8 vertices by pasting these together in a tition function of the Potts model, studied in the- tree-like manner. The paper “Graph Minors XVI” oretical physics in relation to phase transition and shows that a similar structure is obtained if we ex- critical phenomena. The special case when clude any fixed graph M as a minor. Every graph y = −1/x is equivalent to the Jones polynomial of that does not contain M as a minor can be built an alternating link via a projection of the link in from building blocks, each of which is a graph that the plane and medial graph construction. can be “nearly embedded” in some surface in which Proofs of graph minor results are usually hard M cannot be embedded. Different building blocks and technical. This explains why most textbooks intersect each other in bounded size subgraphs and cover only the basic facts about minors. An ex- form a tree-like structure. This is a “rough struc- ception is [1], which is also a good reference for fur- ture result” in the sense that none of its ingredi- ther reading. ents can be eliminated, and in the sense that the structure itself describes a minor-closed family Further Reading such that certain other graphs are excluded from [1] R. DIESTEL, Graph Theory, 3rd Edition, Springer, 2005. this family. [2] N. ROBERTSON and P. D. SEYMOUR, Graph minors.