A characterization of -minor-free graphs
We provide a complete structural characterization of -minor-free graphs. The -connected -minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains nonisomorphic graphs of order for each as well as . To describe the -connected -minor-free graphs we use -outerplanar graphs, graphs embeddable in the plane with a Hamilton -path so that all other edges lie on one side of this path. We show that, subject to an appropriate connectivity condition, -outerplanar graphs are precisely the graphs that have no rooted minor where and correspond to the two vertices on one side of the bipartition of . Each -connected -minor-free graph is then (i) outerplanar, (ii) the union of three -outerplanar graphs and possibly the edge , or (iii) obtained from a -connected -minor-free graph by replacing each edge in a set satisfying a certain condition by an -outerplanar graph. From our characterization it follows that a -minor-free graph has a hamilton cycle if it is -connected and a hamilton path if it is -connected. Also, every -connected -minor-free graph is either planar, or else toroidal and projective-planar.
The Robertson-Seymour Graph Minors project has shown that minor-closed classes of graphs can be described by finitely many forbidden minors. Excluding a small number of minors can give graph classes with interesting properties. The first such result was Wagner’s demonstration  that planar graphs are precisely the graphs that are - and -minor-free.
Excluding certain special classes of graphs as minors seems to give close connections to other graph properties. One of the most important open problems at present is Hadwiger’s Conjecture, which relates excluded complete graph minors to chromatic number. Our interest is in excluding complete bipartite graphs as minors. Together with connectivity conditions, and possibly other assumptions, graphs with no as a minor can be shown to have interesting properties relating to toughness, hamiltonicity, and other traversability properties. The simplest result of this kind follows from a well-known consequence of Wagner’s characterization of planar graphs. This consequence says that -connected -minor-free graphs are outerplanar or ; hence, they are hamiltonian. For some recent examples of this type of result, involving toughness, circumference, and spanning trees of bounded degree, see [1, 2, 13].
Our work was originally motivated by trying to find forbidden minor conditions to make -connected planar graphs, or -connected graphs more generally, hamiltonian. In examining the hamiltonicity of -connected -minor-free graphs we were led to a complete picture of their structure, which we then extended to -minor-free graphs in general. Using this, we show in Section 4 that -connected -minor-free graphs are hamiltonian, and that -connected -minor-free graphs have hamilton paths.
For -minor-free graphs, or -minor-free graphs in general, there are a number of previous results. Dieng and Gavoille (see Dieng’s thesis ) showed that every -connected -minor-free graph contains two vertices whose removal leaves the graph outerplanar. Streib and Young  used Dieng and Gavoille’s result to show that the dimension of the minor poset of a connected graph with no minor is polynomial in . Chen et al.  proved that -connected -minor-free graphs have a cycle of length at least . Myers  proved that a -minor-free graph with satisfies ; more recently Chudnovsky, Reed and Seymour  showed that this is valid for all , and provided stronger bounds for -, - and -connected graphs. Our results improve their bound for -connected graphs when . An unpublished paper of Ding  proposes that -minor-free graphs can be built from slight variations of outerplanar graphs and graphs of bounded order by adding ‘strips’ and ‘fans’ using an operation that is a variant of a -sum (and which corresponds to the idea of replacing subdividable sets of edges that is used later in this paper). Ding’s result involves subgraphs that have minors, and so not all aspects of his structure can be present in the case of -minor-free graphs; our results illuminate the extent to which Ding’s structure still holds.
As part of our work we use rooted minors, where particular vertices of must correspond to certain vertices of when we find as a minor in . For example, Robertson and Seymour  characterized all -connected graphs that have no minor rooted at the three vertices on one side of the bipartition. Fabila-Monroy and Wood  characterized graphs with no minor rooted at all four vertices. Demasi  characterized all -connected planar graphs with no minor rooted at the four vertices on one side of the bipartition. In this paper we characterize all graphs with no minor rooted at two vertices on one side of the bipartition. This result is useful not only here, but also in the authors’ proof that -connected -minor-free planar graphs are hamiltonian (see ).
We begin with some definitions and notation. All graphs are simple. We use ’’ to denote set difference and deletion of vertices from a graph, ’’ to denote deletion of edges, ’’ to denote contraction of edges, and ’’ to denote both addition of edges and join of graphs. Since we work with simple graphs, when we contract an edge any parallel edges formed are reduced to a single edge.
A graph is a minor of a graph if is isomorphic to a graph formed from by contracting and deleting edges of and deleting isolated vertices of . We delete multiple edges and loops, so all minors are simple. Another way to think of a -vertex minor of is as a collection of disjoint subsets of the vertices of , where each corresponds to a vertex of , where (the subgraph of induced by the vertex set ) is connected for , and for each edge there is at least one edge between and in . We call this a model of in . We will often identify minors in graphs by describing the sets . The set is known as the branch set of , and may be thought of as the set of vertices in that contracts to in .
Suppose we are given , , and a bijection . We say that a model of in is a minor rooted at in and at in by if each belongs to the branch set of . If the symmetric group on is a subgroup of the automorphism group of (as it will be in our case) then the exact bijection between and does not matter.
A graph is -minor-free if it does not contain as a minor. A -separation in a graph is a pair of edge-disjoint subgraphs of with , , , and .
Suppose has bipartition . Let and be the branch sets of and in a model of in a graph . Suppose is the branch set of for some . Then there is a path , , with , , and for . Let and let . We can replace with and with and still have a model of (possibly using fewer vertices of than before). Hence without loss of generality we may assume that the branch set of each vertex , , contains a single vertex . Let . We say represents a standard minor. Observe that contains a minor if and only if contains a standard minor. Note that the standard model also applies to minors rooted at two vertices corresponding to and .
A wheel is a graph with . A vertex of degree in is a hub and its incident edges are spokes while the remaining edges form a cycle called the rim. In every vertex is a hub and every edge is both a spoke and a rim edge, but in for there is a unique hub and the edges are partitioned into spokes and rim edges. Note that we identify wheels by their number of vertices, rather than their number of spokes.
A graph is outerplanar if it has an outerplane embedding, an embedding in the plane with every vertex on the outer face.
In the next section, we define a class of graphs and describe several small examples which together make up all -connected -minor-free graphs. We begin with -connected graphs because all -connected graphs on at least six vertices have a minor. This is obvious for complete graphs. Otherwise, a pair of nonadjacent vertices and the four internally disjoint paths between them guaranteed by Menger’s Theorem yield a minor. In Section 3 we extend the characterization to -connected graphs. The generalization to all graphs follows because a graph that is not -connected is -minor-free if and only if each of its blocks is -minor-free. Section 4 presents applications of our characterization to hamiltonicity, topological properties, counting, and edge bounds.
2 The -connected case
All graphs with are trivially -minor-free; the -connected ones are , , , and . For , first we define a class of graphs and identify those that are -connected and -minor-free. We then look at some small graphs that do not fit into this class. Finally, we show that every -connected -minor-free graph is one of these we have described.
2.1 A class of graphs
For and , let consist of a spanning path , which we call the spine, and edges for and for . The graph is ; we call the plus edge. All graphs are planar. The graph is a wheel with hub . Examples are shown in Figure 1. Since we often assume .
In the following three lemmas we first determine when a graph is -connected, and then when it is -minor-free.
For , is -connected if and only if (i) , , and the plus edge is present (or symmetrically , , and the plus edge is present) or (ii) and .
Assume that . To prove the forward direction, assume is -connected and first suppose . If the plus edge is not present, then has degree and is a -cut. Similarly if , then has degree and is a -cut. Next suppose . If , then there is necessarily a degree vertex with and hence a -cut in .
To prove the reverse direction, assume (i) or (ii). If (i) holds, is a wheel, which is -connected, so we may assume that (ii) holds. To show -connectedness we find three internally disjoint paths between each possible pair of vertices. For and we have paths , , and (where possibly ). Next suppose that only one of and is in the considered pair, say without loss of generality. First consider and where . When , then the three disjoint paths are , , and . When , then and and the three disjoint paths are , , and . Now consider and where . Then the three disjoint paths are , , and . Finally consider and where and . If and are both adjacent to the same end vertex, say , where , then the three disjoint paths are , , and . Otherwise the three disjoint paths are , , and . ∎
For , is -minor-free if and only if .
To prove the forward direction, suppose . Then there are vertices and such that both and are adjacent to both and and . Then there is a standard minor in : let , , and .
Now suppose that . We claim that if has a standard minor , then and (or vice versa). The graph is outerplanar and thus has no minor. Therefore, if has a minor, then it must include . We cannot have because then the outerplanar graph would have a minor. By symmetry, must also be included in the minor and . If , then has a minor, but is a path and there is no minor in a path. The only remaining possibility is and (or vice versa).
Let denote the set of neighbors of . Let and , which intersect only if . Suppose has a standard minor . Then by the claim proved in the previous paragraph, and . We consider the makeup of . Suppose , in that order along the spine. Since separates and , and , we cannot have adjacent to , which is a contradiction. Thus . Symmetrically, . We must have and in the order along the spine. Since , there must be a -path in , and hence . Then is a cutvertex separating and in , so . Now there must also be a -path in but no such path exists. Thus there is no minor. ∎
Define to be the set of (labeled) graphs of the form that are both -connected and -minor-free. Of the four -connected graphs on fewer than six vertices, three are planar, and all three belong to : , , and . From this and Lemmas 2.1 and 2.2 we get
Let denote the class of all graphs isomorphic to a graph in . Note that graphs in are -sums of two wheels, a fact we will see in more detail later on.
There are some isomorphisms between graphs in and also symmetries within certain graphs of the class. Let be the involution with for . Then provides the isomorphism (in both directions) between and that we have already noted; if it is an automorphism. The graph is isomorphic to , with as a hub. It has the obvious symmetries.
Define to be the involution fixing and and with for . Then is an automorphism of , an isomorphism (in both directions) between and , and an automorphism of . The case is illustrated in Figure 2, where corresponds to reflection about a vertical axis. The graph without the dashed edges and is . With the edge , the graph is and with , the graph is . With both edges and , the graph is . In general maps the spine to the path . For with we call the second spine. When we have a similar involution , and the path can be regarded as an extra spine. When , is the image of under the automorphism .
Finally, besides some obvious special symmetries when or , is vertex-transitive and is isomorphic to the triangular prism.
These symmetries and isomorphisms will be important later, particularly in Section 3 when we discuss which edges of can be subdivided without creating a minor. Up to isomorphism the class contains one -vertex graph and -vertex graphs for each .
We now examine the effect of deleting or contracting a single edge of a graph in .
Suppose and . The following are equivalent.
(ii) is -connected.
(iii) is not a wheel and either is a plus edge, or and .
Clearly (iii) (i) (ii). If (iii) does not hold then has at least one vertex of degree , so (ii) does not hold; thus (ii) (iii). ∎
|isomorphic to||is -conn.?||?|
|,||if plus edge||if plus edge|
Now consider contracting an edge of . If then and for any edge , so assume that . If is a wheel then we obtain if we contract a rim edge, and if we contract a spoke. Therefore assume is not a wheel, so . The effects of contracting edges in this case are shown in Table 1. Here the superscript ’’ means that the plus edge is present in precisely if it is present in . Edges not included in the table are covered by the symmetry that swaps and , and . We may summarize the results as follows.
Suppose and .
(i) If then is isomorphic to a graph in with at most one edge deleted.
(ii) If then if and only if is -connected.
(iii) If is a wheel with then some is isomorphic to , and every is isomorphic to . If is not a wheel then (from the starred entries in Table 1) some is isomorphic to each of , and, if , also ; and any is isomorphic to a spanning subgraph of one of these.
Now we apply these results to the structure of minors of graphs in or .
Every minor of a graph in is a subgraph of some graph in .
Apply Lemma 2.4(i) repeatedly to replace contractions by deletions (details are left to the reader). ∎
If a -connected graph is a minor of a -connected graph , then there is a sequence of -connected graphs where , , and each is obtained from by contraction or deletion of a single edge.
Seymour’s Splitter Theorem  as applied to graphs, or a similar result of Negami , says that our result is true if is not a wheel, or if is the largest wheel minor of . Seymour’s operations and connectivity are defined for graphs with loops and multiple edges, not simple graphs, which is why a sequence of minors cannot be continued to reduce a large wheel minor to a smaller one. In particular, contracting a rim edge of a wheel in his definition yields a pair of parallel edges and so by his definition the graph is not -connected. With our definition, where we reduce parallel edges to a single edge after contraction, we can contract a rim edge of a wheel , , to obtain the smaller wheel , which is still -connected. Therefore, we can continue the sequence of operations to also reach wheel minors that are not the largest wheel minor. ∎
If is a -connected minor of then .
2.2 Small cases
Figure 3 shows nine small graphs that are -connected (easily checked), not in (also easily checked; all but have a minor and so are nonplanar) and -minor-free. The first graph, , is the only -connected graph on fewer than six vertices that is not in . To prove that the other eight graphs are -minor-free we examine the two maximal graphs and , and show that the rest are minors of .
The graph is -minor-free.
Consider with vertices labeled as on the left in Figure 4. Suppose there is a standard minor in and suppose . Then must be either or since these are the only vertices of degree . Say, without loss of generality, . Then , and must be a subset of . None of these three vertices are adjacent to , however, so we cannot have adjacent to and thus we cannot have , or symmetrically . Thus and and since , .
Let be a triangle with a set of neighbors with . Suppose . Then we would have along with the third vertex of , but separates from the rest of the graph so cannot be adjacent to . Thus (or symmetrically ) cannot consist of two vertices in a triangle with only three neighbors. In , we have the following triples of vertices which form such triangles: , , , and . The only remaining pairs of adjacent vertices that could make up or are , , and where all three cases are symmetric. If , then must be but this set is not an option for . ∎
The graph is -minor-free.
It is easy to check that has no subgraph isomorphic to , nor does for any . Hence is -minor-free since . ∎
Figure 5 shows what we will prove is the Hasse diagram for the minor ordering of all -connected minors of (also labeled , following Ding and Liu ) and . For future reference the figure also includes three additional, circled graphs (the cube), (contract any edge of ) and (the -vertex twisted cube or Möbius ladder). Unlike the other graphs, these three have minors, as shown by the minor in on the right in Figure 6. Here, and later, a minor is indicated by two groups of vertices circled by dotted curves representing the two vertices in one part of the bipartition of , and four triangular vertices representing the other part.
Figure 5 is the Hasse diagram for all -connected minors (up to isomorphism) of , , and .
By Lemma 2.6, we can proceed by single edge deletions and contractions, and we do not need to consider further minors once we reach a graph that is not -connected. The figure is clearly correct for the -connected graphs on four or five vertices, so we consider only graphs with at least six vertices. Also, the -connected minors for graphs in follow from Lemmas 2.3 and 2.4(iii), so we consider only graphs not in .
In what follows results of all deletions or contractions are identified only up to isomorphism. When we lose -connectivity, in all but one case there will be at least one vertex of degree . We work upwards in the figure.
For the graphs , and label the vertices consecutively along the top row then the bottom row in Figure 3. For , deleting any edge loses -connectivity; contracting any edge results in . For , deleting yields , and deleting any other edge loses -connectivity. Contracting an edge incident with yields , contracting loses -connectivity, and contracting any other edge incident with or yields . For , all edges are equivalent up to symmetry to one of , , or . Deleting or loses -connectivity, deleting gives , and deleting gives . Contracting gives , contracting gives , contracting loses -connectivity, and contracting yields .
For and we redraw as on the left in Figure 6 and take . For , deleting any edge loses -connectivity. Up to symmetry, there are five edge contractions to consider: , , , and . Contracting yields , contracting loses -connectivity, contracting yields , contracting results in , and contracting gives . For , all edges are equivalent up to symmetry to six possibilities: , , , , and . Deleting , , or loses -connectivity, deleting yields , and deleting results in . Contracting or loses -connectivity, contracting results in , contracting yields , contracting gives , and contracting gives .
For we label the vertices as on the left in Figure 4 and take . Deleting any edge of loses -connectivity. Up to symmetry, there are three edge contractions to consider: , and . Contracting loses -connectivity, contracting results in , and contracting yields . For , deleting yields and deleting any other edge loses -connectivity. Up to symmetry, there are four edge contractions of to consider: , , and . Contracting or loses -connectivity, contracting results in , and contracting gives .
We label as on the right in Figure 4. Up to symmetry all edges are equivalent to one of four edges: , , and . Deleting results in , and deleting any of the other three edges loses -connectivity. Contracting or loses -connectivity, contracting yields the triangular prism , and contracting results in .
Finally, consider , and . Label as shown on the right in Figure 6. Every edge in is adjacent to a degree vertex so deleting any edge loses -connectivity. Up to symmetry, there are four edge contractions to consider: , , and . Contracting loses -connectivity, and contracting also loses -connectivity (without creating a vertex of degree ). Contracting results in , and contracting yields . In the cube all edges are symmetric; deleting any edge loses -connectivity, and contracting any edge yields . We may take to be with added diagonals for . Deleting any edge loses -connectivity, contracting a edge results in , and contracting a diagonal yields . ∎
Considering the minors of , we obtain the following.
The graphs , , , , , and are -minor-free.
2.3 Characterization of -connected graphs
Let be a -connected graph. Then is -minor-free if and only if or is isomorphic to one of the nine small exceptions shown in Figure 3.
Our original proof of this theorem examined the structure of a -connected -minor-free graph relative to a longest non-hamilton cycle in the graph. We analyzed cases and either derived a contradiction with a longer non-hamilton cycle or a minor, or found a desired graph. However, we then discovered the recent systematic investigation by Ding and Liu , characterizing -minor-free graphs for all -connected graphs on at most eleven edges. These allow us to give a shorter proof, which we present here.
First we give some definitions. Denote by Oct the graph obtained from the octahedron by removing one edge. A -sum of two -connected graphs and is a graph obtained by identifying a triangle of with a triangle of and possibly deleting some of the edges of the common triangle as long as no degree vertices are created. Any -cut in would lead to a -cut in either or so is -connected. An example is the graph which is a -sum of and a triangular prism. A common -sum of three or more graphs is formed by specifying one triangle in each graph and identifying all as a single triangle called the common triangle; again edges of the common triangle may be deleted as long as no degree vertices are created. Let be the set of all graphs formed by taking common -sums of wheels and triangular prisms. All graphs in are -connected. We use the following result due to Ding and Liu.
Theorem 2.13 (Ding and Liu ).
Up to isomorphism the family of -connected Oct-minor-free graphs consists of graphs in and -connected minors of , , and .