The structure of graphs not topologically containing the Wagner graph John Maharry, Neil Robertson1 Department of Mathematics The Ohio State University Columbus, OH USA [email protected], [email protected] Abstract A structural characterization of graphs not containing the Wagner graph, also known as V8, is shown. The result was announced in 1979 by the second author, but until now a proof has not been published. Keywords: Wagner graph, V8 graph, Excluded-minor Graphs in this paper are finite and simple in the sense that they have no loops or multiple edges. Such graphs are determined by their vertex- adjacency relation. The Wagner graph is the graph W which is formed by adding to an octagon four edges joining its diagonally opposite pairs of vertices. This graph appears in a theorem of Wagner [18] which states that any graph with no minor isomorphic to K5 can be obtained by 0; 1; 2 and 3-summing beginning with planar graphs and the graph W . The graph W is also known as V8 or the M¨obiusladder with four rungs. Given a graph G, one can obtain a contraction K of G by contracting pairwise disjoint connected induced subgraphs to single (distinct) vertices where distinct vertices of K are adjacent if and only if there exists an edge of G with an endvertex in each of the corresponding subgraphs of G. A graph M is a minor of a graph G when M is a contraction of a subgraph of G. We write H ≤m G when H is isomorphic to a minor M of G. Given an edge e 2 E(G) with endvertices x and y, we define the single-edge contraction 1Research funded in part by King Abdulaziz University, 2011 and in part by SERC Visiting Fellowship Research Grant, University of London, 1985 Preprint submitted to Journal Combin. Ser. B October 4, 2013 Figure 1: The Wagner Graph W , also known as V8 to be the graph formed by contracting the edge e to a new vertex, call it z. Formally, the result of the single-edge contraction is the graph G=e with vertex set V (G=e) := (V (G) n fx; yg) [ fzg and edge set E(G=e) := fuv 2 E(G)jfu; vg \ fx; yg = ;g [ fzwjxw 2 E(G) n feg or yw 2 E(G) n fegg. An equivalent definition of minor inclusion of a graph is H ≤m G if and only if H is isomorphic to a graph obtained by a series of single-edge contractions starting with a subgraph S of G. A single-edge contraction of an edge with at least one end-vertex divalent is called a series contraction. If one restricts the single-edge contractions to series-contractions, then the resulting graph H is a topological minor of G. In this case, the subgraph S of G is called a subdivision of H in G. We write H ≤t G when H is isomorphic to a topological minor of G. In such a topological inclusion, the edges of H correspond to paths in S whose internal vertices are divalent in S, which we call branches. Similarly, the vertices of S that correspond to vertices of H are called nodes. Note that if the edge e is contained in a triangle T of S then the two edges of E(T ) n feg become one edge of S=e. Otherwise, when no such (single-edge) contraction is made, the resulting graph S=e is topologically equivalent to S, i.e. the underlying topological representations of S and S=e are homeomorphic. Note that Wagner's theorem implies that W is the unique maximal 3- connected graph which does not contain K5 as a minor. There are no maximal 3-connected planar graphs under the relation of minor inclusion. In a sense, the graph W can be viewed as a graph intermediate between the Kuratowski graph K3;3 and the Petersen graph P (the unique 3-regular graph of girth 5 on 10 vertices). In fact the chain K3;3 <t W <t P under topological inclusion cannot be refined to a longer chain. It is straightforward to check that as W is a cubic graph, containing W as a minor is equivalent to containing W as 2 a topological subgraph. In this paper, we present an exact characterization of those graphs which do not contain a subgraph topologically equivalent to W . This theorem was originally announced in 1979 by the second author. It has become widely known. However no proof has, until now, ever appeared in print. As the theorem has recently been used as a starting point for several other theorems, it is time the result is written formally. The proof in this paper has been streamlined and modified and is based on a partial hand-written draft by the second author from 1985. The result is central to the proofs of at least four recent results: a characterization of the flexibility of graph embeddings on the Projective Plane [10], a characterization of 2-crossing critical graphs [1], a structural characterization of graphs with no Octahedron minor by Ding [4] and a paper showing that all V8-free graphs have a closed 2-cell embedding [13]. Prior to the initial announcement of this result, there were few well-known excluded minor characterizations. The best known were results for K5, K3;3 and the 3-Prism [18, 19, 3]. More recently, a number of other graphs have been so characterized, including the Cube [9], the Octahedron [4], the 5- and 6-wheels [6, 7] and several graphs on ten or fewer edges [5]. 1. Introduction Consider the problem of finding a subdivision of W contained in a graph G, i.e. concretely showing that W ≤t G. First, by a standard argument we can reduce to the case where G is `internally-4-connected'. Define a separation of G to be an expression G = G1 [ G2, where G1, G2 are non-null edge-disjoint subgraphs of G and jE(G1)j ≥ jV (G1 \ G2)j ≤ jE(G2)j. Suppose jV (G1 \ G2)j ≤ 3. When jV (G1 \ G2)j ≤ 1 it is clear that W ≤t G if and only if W ≤t G1 or W ≤t G2. If such separations do not occur and jV (G)j ≥ 3 then G is said to be 2-connected. Assume j \ j + + this and suppose V (G1 G2) = 2. Define G1 and G2 to be G1 and G2, respectively, with an edge added where necessary joining the two common + + ≤ ≤ vertices. Then G1 , G2 t G and it is easy to see that W t G if and ≤ + ≤ + only if W t G1 or W t G2 . Again, if such separations do not occur and jV (G)j ≥ 4 then G is 3-connected. Assume this and suppose jV (G1 \G2)j = 3 j j ≥ ≤ j j + + and E(G1) 4 E(G2) . Define G1 and G2 to be G1 and G2, with new vertices x1 and x2 respectively, adjacent to each of the three vertices of G1\G2 + + ≤ ≤ by new edges. Again, the reader can verify that G1 , G2 t G, and W t G 3 ≤ + ≤ + if and only if W t G1 or W t G2 . Note also, for inductive purposes, that + + G1;G2 are proper subgraphs of G and that G1 and G2 are both properly topologically contained in G. When a graph G does not admit separations of the type described above and jV (G)j ≥ 4, it is said to be internally-4-connected. It is readily seen that W itself is internally-4-connected and that K4 is the smallest such graph. We can also see that all the above remarks go through with similar proofs when W is replaced by the graph C of the 3-dimensional cube. Moreover, when an internally-4-connected graph G contains disjoint circuits Q1 and Q2, each of length at least 4, then Menger's theorem can be applied to produce a 4-join from Q1 to Q2, i.e. four vertex-disjoint simple paths from Q1 to Q2. These can be seen (Cor. 2.6) to produce either W or C embedded in G. It is thus of interest to characterize the internally-4-connected graphs which do not contain disjoint circuits of length ≥ 4. Note that if a vertex is both in a triangle and of valency three, this violates the conditions of internal-4-connection except in the case of K4. Proposition 1.1. If G is an internally-4{connected graph then either G has disjoint cicuits Q , Q with jE(Q )j ≥ 4 ≤ jE(Q )j, or jV (G)j ≤ 7, or ∼ 1 2 1 2 G = L(K3;3). Proof of this proposition is postponed until the main result of the paper is stated. However, by our earlier remarks, it shows that if G is not too small and is not isomorphic to one particular exceptional graph, L(K3;3), the line graph of K3;3, then it contains either W or C as an embedded subgraph. It may be noted that there are a fairly large number of `small' internally-4-connected graphs G. The only internally-4-connected graphs on six or fewer vertices are K4,K5,K3;3 and the Octahedron with k = 0; 1; 2; 3 diagonals added. On seven vertices, one can verify that there are exactly 28 internally-4-connected graphs. When V = 8 we have either W or C as a spanning subgraph, by the above argument, and so there are just the many supergraphs of these which do not contain an incident triad-triangle pair in the class of graphs.