Sublinear separators, fragility and subexponential expansion

Let G be a subgraph-closed graph class with bounded maximum degree. We show that if G has balanced separators whose size is smaller than linear by a polynomial factor, then G has subexponential expansion. This gives a partial converse to a result of Nešetřil and Ossona de Mendez. As an intermediate step, the proof uses a new kind of graph decompositions. The concept of graph classes with bounded expansion was introduced by Nešetřil and Ossona de Mendez [31] as a way of formalizing the notion of sparse graph classes. Let us give a few definitions. For a graph G, a k-minor of G is any graph obtained from G by contracting pairwise vertex-disjoint subgraphs of radius at most k and removing vertices and edges. Thus, a 0-minor is just a subgraph of G. Let us define ∇k(G) as max { |E(G′)| |V (G′)| : G′ is a k-minor of G }