5 problems

Let $\mathcal{G}$ be a graph class with bounded expansion, let $I$ be a simple first-order graph interpretation scheme, and let $\varphi$ be a first-order property to be tested. Th…

Let $\mathcal{C}$ be a monotone class of graphs. For each fixed $h\geq 1$, consider posets of height at most $h$ whose cover graphs belong to $\mathcal{C}$. Bounded-expansion chara…

Bounded-expansion characterization. The class $\mathcal{C}$ has bounded expansion if and only if, for each $h\geq 1$, the posets of height $h$ whose cover graphs are in…

Structural conjecture for nowhere dense classes. If $\mathcal C$ does not have bounded expansion, then there exists an integer $p$ such that $\mathcal C$ includes $p$-subdivisions…

Bounded expansion characterization. The following properties are equivalent: for every integer $p$ there is a non-bipartite graph $H_p$ of odd-girth strictly greater than $2p+1$ an…