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The Structure of Graphs Not Topologically Containing the Wagner Graph

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The Structure of Graphs Not Topologically Containing the Wagner Graph 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 1. Introduction Graphs in this paper are finite and have no loops or multiple edges. Given a graph G, one can obtain a contraction K of G by contracting pairwise dis- joint 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 end- vertex 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 to be the graph formed by contracting the edge e to a new vertex, call it z. An equivalent definition of minor inclusion of a graph is H ≤m G if and only if H is iso- morphic to a graph obtained by a series of single-edge contractions starting with a subgraph S of G. 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 December 3, 2015 A single-edge contraction of an edge with at least one endvertex 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. The corresponding paths in S are called branches. Similarly, the vertices of S that correspond to vertices of H are called nodes. Note that H is isomorphic to the graph of nodes and branches of S we will normally denote this graph by H and S by H0. The subpath of a branch M from vertex a to vertex b on M will be denoted M[a; b], or simply [a; b]. We will use standard notation such as (a; b) if one or both of the endpoints are not included. 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. A separation where jV (G1 \ G2)j = k is called a k-separation.A graph G is k-connected if G has no k0-separation for any k0 < k and jV (G)j > k. A 3-connected graph G is called internally-4-connected if jV (G)j ≥ 4 and there does not exist a 3-separation G = G1[G2 where jE(G1)j ≥ 4 ≤ jE(G2)j. Note that, unless G is isomorphic to K4, if G contains a vertex that is both in a triangle and of valency three, then G is not internally-4-connected. 2. Main Theorem 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 [20] 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 (see Figure 1) is also known as V8 or the M¨obiusladder with four rungs. Note that Wagner's theorem implies that W is the unique maximal 3-connected nonplanar graph that does not contain K5 as a minor. 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). It is straightforward to check that as W is a cubic graph, containing W as a minor is equivalent to containing W under topological inclusion. In this paper, we present an exact characterization of those internally-4- connected graphs that do not contain as a subgraph a subdivision W . This 2 Figure 1: The Wagner Graph W , also known as V8 theorem was originally announced in 1979 by the second author. It has become widely known. However no proof has, until now, 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 is based on a partial hand-written draft by the second author from 1985 and has been streamlined and modified. The result is central to the proofs of at least four recent results: a characterization of 2-crossing critical graphs [1], a structural characterization of graphs with no Octahedron minor [4], a characterization of the flexibility of graph embeddings on the Projective Plane [12] and a paper showing that all V8-free graphs have a closed 2-cell embedding [15]. 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 [20, 21, 3]. More recently, a number of other graphs have been so characterized, including the Cube [11], the Octahedron [4], the complement of P7 [6], the 5- and 6-wheels [7, 8] and several graphs on ten or fewer edges [5]. We now state the Main Theorem of this paper. Here L(K3;3) denotes the line graph of K3;3. Theorem 2.1. If G is an internally-4-connected graph then either G con- tains a subgraph S which is isomorphic to a subdivision of W or one of the following structural conditions holds: 1. G is planar, 2. G − fx; yg is a cycle, for some adjacent x; y 2 V (G), 3. G − fw; x; y; zg is edgeless, for some w; x; y; z 2 V (G), ∼ 4. G = L(K3;3), or 3 5. jV (G)j ≤ 7. Suppose G is an internally-4-connected graph. If vertices x; y exist such that G − fx; yg is a cycle as in Case 2, then G is called a double wheel. We refer to the rim, hub and spokes of G naturally. We observe that double wheels are planar or not planar according as x; y are nonadjacent or adjacent. It is routine to check that in an internally-4-connected double wheel G, the hub vertices x and y are either both adjacent to all vertices of the rim or to successively alternating vertices of an even length rim. Such double wheels are called complete or alternating, respectively. When nonplanar, complete double wheels exist if jV (G)j ≥ 5 and alternating double wheels exist for even jV (G)j ≥ 8, as internal-4-connectivity fails at lower values. When jV (G)j ≥ 8, a double wheel contains a topological Cube C0 with its rim C0 − fx; yg subdivided and no other edges subdivided. Note that x and y are antipodal nodes of C0. If vertices w; x; y; z exist such that G − fw; x; y; zg is edgeless as in Case 3, then G is called vertex 4-covered and fw; x; y; zg is a vertex 4-cover of G. When jV (G)j ≥ 8, a vertex 4-covered graph contains a Cube C as a subgraph with fw; x; y; zg as a colour class. Conversely, if G contains a subdivided Cube C0, then C = C0 and a vertex 4-cover fw; x; y; zg must be a colour class of C. Proposition 2.2. W is not contained topologically in G when the conditions of 2.1 hold. Proof: To show this, suppose for a contradiction that S which is isomor- phic to a subdivision of W , is a subgraph of G. For case (1), W , and hence S, is nonplanar. For case (2), if S is a subgraph of G, then S − fx; yg is a subgraph of G − fx; yg, which is a cycle. Hence, the valencies of the (at least) six remaining nodes of S were reduced when x and y were removed. This could only occur if x and y are nodes of S which are adjacent to the other 6 nodes of S. As W has diameter 2, this cannot happen for S. For case (3), note that the edge covering number of a subdivision is at least the edge covering number of the original graph. Since the edge covering number of W is five, S − fw; x; y; zg cannot be edgeless. For case (4), L(K3;3) does not have disjoint cycles of length at least 4, because the only quadrilateral in L(K3;3), up to isomorphism, leaves two triangles with one common vertex in its (vertex-set) complement. Finally, for case (5), the graphs are too small. 4 Note that the only internally-4-connected graphs on six or fewer vertices are K4,K5,K3;3 and the Octahedron with 0; 1; 2; or 3 diagonals added.
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