More generally, a
forbidden graph characterization is a method of
specifying a family of
graph, or
hypergraph, structures, by specifying substructures that are forbidden to exist within any graph in the family. Different families vary in the nature of what is
forbidden. In general, a structure
G is a member of a family \mathcal{F}
if and only if a forbidden substructure is
not contained in
G. The
forbidden substructure might be one of: •
subgraphs, smaller graphs obtained from subsets of the vertices and edges of a larger graph, •
induced subgraphs, smaller graphs obtained by selecting a subset of the vertices and using all edges with both endpoints in that subset, •
homeomorphic subgraphs (also called
topological minors), smaller graphs obtained from subgraphs by collapsing paths of degree-two vertices to single edges, or •
graph minors, smaller graphs obtained from subgraphs by arbitrary
edge contractions. The set of structures that are forbidden from belonging to a given graph family can also be called an
obstruction set for that family. Forbidden graph characterizations may be used in
algorithms for testing whether a graph belongs to a given family. In many cases, it is possible to test in
polynomial time whether a given graph contains any of the members of the obstruction set, and therefore whether it belongs to the family defined by that obstruction set. In order for a family to have a forbidden graph characterization, with a particular type of substructure, the family must be closed under substructures. That is, every substructure (of a given type) of a graph in the family must be another graph in the family. Equivalently, if a graph is not part of the family, all larger graphs containing it as a substructure must also be excluded from the family. When this is true, there always exists an obstruction set (the set of graphs that are not in the family but whose smaller substructures all belong to the family). However, for some notions of what a substructure is, this obstruction set could be infinite. The
Robertson–Seymour theorem proves that, for the particular case of
graph minors, a family that is closed under minors always has a finite obstruction set. ==List of forbidden characterizations for graphs and hypergraphs==