**Examples of Minor-closed Families**

The following sets of finite graphs are minor-closed, and therefore (by the Robertson–Seymour theorem) have forbidden minor characterizations:

- forests, linear forests (disjoint unions of path graphs), pseudoforests, and cactus graphs;
- planar graphs, outerplanar graphs, apex graphs (formed by adding a single vertex to a planar graph), toroidal graphs, and the graphs that can be embedded on any fixed two-dimensional manifold;
- graphs that are linklessly embeddable in Euclidean 3-space, and graphs that are knotlessly embeddable in Euclidean 3-space;
- graphs with a feedback vertex set of size bounded by some fixed constant; graphs with Colin de Verdière graph invariant bounded by some fixed constant; graphs with treewidth, pathwidth, or branchwidth bounded by some fixed constant.

Read more about this topic: Robertson–Seymour Theorem

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