Many natural families of sparse graphs have bounded expansion.
A closely related but stronger property, polynomial expansion, is equivalent to the existence of separator theorems for these families.
Families with these properties have efficient algorithms for problems including the subgraph isomorphism problem and model checking for the first order theory of graphs.
Conversely, graphs with polynomial expansion have sublinear separator theorems.
[4][5][6][7] In higher dimensional Euclidean spaces, intersection graphs of systems of balls with the property that any point of space is covered by a bounded number of balls also obey separator theorems[8] that imply polynomial expansion.