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Unformatted text preview: Computational Topology (Jeff Erickson) Graph Minors Wagner did indeed discuss this problem in the 1960s with his then students, Halin and Mader, and it is not unthinkable that one of them conjectured a positive solution. Wagner himself always insisted that he did not—even after the graph minor theorem had been proved. — Reinhard Diestel, Graph Theory , 3rd edition (2005) Unfortunately, for any instance G =( V , E ) that one could fit into the known universe, one would easily prefer | V | 70 to even constant time, if that constant had to be one of Robertson and Seymour’s. — David Johnson, “The NP-completeness Column” (1987) 12 Graph Minors A minor of a graph G is a graph obtained from G by contracting edges, deleting edges, and deleting isolated vertices; a proper minor of G is any minor other than G itself. For example, the complete graph K 5 and the complete bipartite graph K 3,3 are both minors of the infamous Peterson graph: Both K 5 and K 3,3 are minors of the Peterson graph. Doubled edges are contracted; dashed edges are deleted. A classical theorem of Kuratowski [ 21 ] states that a graph is planar if and only if it does not contain a subdivision of K 5 or K 3,3 as a subgraph. Kuratowski’s theorem was refined by Wagner in his 1935 PhD thesis: Theorem 12.1 (Wagner [ 43 ] ). A graph G is planar if and only if K 5 and K 3,3 are not minors of G . Wagner’s thesis continued with a characterization of all graphs that do not have K 5 (but may have K 3,3 ) as a minor [ 42 ] . Wagner’s work led to a more general study of families of graphs with forbidden minors . A graph H is a forbidden minor for a set F of graphs if H is not a minor of any graph in F . A forbidden minor H of F is minimal if no proper minor of H is also a forbidden minor. It’s quite easy to see that any family F of graphs with at least one forbidden minor is minor-closed : Every minor of a graph in F is also in F . Conversely, if a minor-closed family of graphs excludes a graph H , it must also exclude any graph for which H is a minor. Thus, every minor-closed family of graphs, except the family of all graphs, has at least one forbidden minor. In the mid-1980s, Neil Robertson and Paul Seymour announced a proof of one of the deepest theorems in combinatorics [ 25 ] ; the details of their proof were published over the next two decades in a series of 21 papers totalling several hundred pages. According to Robertson and Seymour, this theorem was conjectured by Wagner as early as the 1930s, although his conjecture did not appear in print until many decades later [ 44 ] . 1 Computational Topology (Jeff Erickson) Graph Minors The Graph Minor Theorem (Robertson and Seymour [ 29 ] ). In any infinite set of graphs, at least one graph is a proper minor of another....
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- Fall '08
- Graph Theory, Planar graph, Neil Robertson, K. Wagner