Graphs with Geometric Properties

There are many graphs with the property that every subgraph of a given simple isomorphism type can be completed to a larger subgraph which is embedded in its ambient parent graph in a nice way. Often, such graphs can be classified up to isomorphism. Here we survey theorems on polar space graphs, graphs with the cotriangle property, copolar graphs, Fischer spaces, and generalized Fischer spaces, as well as graphs with the odd coclique property.