Extremal functions for sparse minors

The notion of a graph minor, which generalizes graph subgraphs, is a central notion of modern graph theory. Classical results concerning graph minors include the Graph Minor Theorem and the Graph Structure Theorem, both due to Robertson and Seymour. The results concern properties of classes of graphs closed under taking minors; such graph classes include many important natural classes of graphs, e.g., the class of planar graphs and, more generally, the class of graphs embeddable in a fixed surface. The Graph Minor Theorem asserts that every class of graphs closed under taking minors has a finite list of forbidden minors. For example, Wagner’s Theorem, which claims that a graph is planar if and only if it does not contain or as a minor, is a particular case of this theorem. The Graph Structure Theorem asserts that graphs from a fixed class of graphs closed under taking minors can be decomposed in a tree-like fashion into graphs almost embeddable in a fixed surface. In particular, every graph in a class of graphs avoiding a fixed minor admits strongly sublinear separators (the Planar separator theorem of Lipton and Tarjan is a special case of this more general result). As the number of edges of every graph contained in a class of graphs closed under taking minors is linear in the number of its vertices, one can define to be the maximum possible density of a graph that does not contain a graph as a minor. This quantity has been a subject of very intensive research; for example, a long list of bounds concerning culminated with a result of Thomason in 2001, who precisely determined its asymptotic behavior. This paper provides bounds on when itself is from a class of sparse graphs. In particular, the authors prove an asymptotically tight bound on in terms of the number of vertices of and the ratio of the vertex cover and the number of vertices of graphs contained in a class of graphs with strongly sublinear separators.