In
graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor. A prototypical example of this phenomenon is
Kuratowski's theorem, which states that a graph is
planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the
complete graph K5 and the
complete bipartite graph K3,3. For Kuratowski's theorem, the notion of containment is that of
graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of one of these two graphs as a subgraph (in which case it does not belong to the planar graphs).