Minor-closed Graph Families
For more details on minor-closed graph families, including a list of some, see Robertson–Seymour theorem.Many families of graphs have the property that every minor of a graph in F is also in F; such a class is said to be minor-closed. For instance, in any planar graph, or any embedding of a graph on a fixed topological surface, neither the removal of edges nor the contraction of edges can increase the genus of the embedding; therefore, planar graphs and the graphs embeddable on any fixed surface form minor-closed families.
If F is a minor-closed family, then (because of the well-quasi-ordering property of minors) among the graphs that do not belong to F there is a finite set X of minor-minimal graphs. These graphs are forbidden minors for F: a graph belongs to F if and only if it does not contain as a minor any graph in X. That is, every minor-closed family F can be characterized as the family of X-minor-free graphs for some finite set X of forbidden minors. The best-known example of a characterization of this type is Wagner's theorem characterizing the planar graphs as the graphs having neither K5 nor K3,3 as minors.
In some cases, the properties of the graphs in a minor-closed family may be closely connected to the properties of their excluded minors. For example a minor-closed graph family F has bounded pathwidth if and only if its forbidden minors include a forest, F has bounded tree-depth if and only if its forbidden minors include a disjoint union of path graphs, F has bounded treewidth if and only if its forbidden minors include a planar graph, and F has bounded local treewidth (a functional relationship between diameter and treewidth) if and only if its forbidden minors include an apex graph (a graph that can be made planar by the removal of a single vertex). If H can be drawn in the plane with only a single crossing (that is, it has crossing number one) then the H-minor-free graphs have a simplified structure theorem in which they are formed as clique-sums of planar graphs and graphs of bounded treewidth. For instance, both K5 and K3,3 have crossing number one, and as Wagner showed the K5-free graphs are exactly the 3-clique-sums of planar graphs and the eight-vertex Wagner graph, while the K3,3-free graphs are exactly the 2-clique-sums of planar graphs and K5.
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