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,, Theorem 9, p. 81; . 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.; ; .
