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Robertson–Seymour theorem
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==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: *[[forest (graph theory)|forests]], linear forests ([[disjoint union]]s of [[path graph]]s), [[pseudoforest]]s, and [[cactus graph]]s; *[[planar graph]]s, [[outerplanar graph]]s, [[apex graph]]s (formed by adding a single vertex to a planar graph), [[toroidal graph]]s, and the graphs that can be [[graph embedding|embedded]] on any fixed two-dimensional [[manifold]];<ref name="l05-76-77">{{harvtxt|Lovász|2005|pp=76–77}}.</ref> *graphs that are [[linkless embedding|linklessly embeddable]] in Euclidean 3-space, and graphs that are [[knot (mathematics)|knotlessly]] embeddable in Euclidean 3-space;<ref name="l05-76-77"/> *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.
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