13 problems

Let $\mathcal{D}$ be a graph class, and for every FO formula $\psi(x,y)$ let $F_\psi$ be a graph such that $F_\psi$ is not an induced subgraph of any member of…

Stable transduction conjecture. A hereditary class of graphs is stable if and only if it is a transduction of a nowhere dense class of graphs.

Nešetřil–Ossona de Mendez's modeling-limit conjecture. Every first-order convergent sequence of graphs from a nowhere-dense class of graphs has a modeling limit.

Let $\mathcal{C}$ be a nowhere dense class of graphs, let $d$ be a positive integer, and let $\varepsilon$ be a positive real. For a graph $G$, write $G^d$ for its $d$th power, and…

Let $G$ be a graph from a nowhere dense class, and let $k$ be the parameter bounding the size of the connected dominating sets in the token-jumping connected dominating set reconfi…

Stable hereditary graph class characterization conjecture. The following properties are equivalent:

Monadic stability characterization conjecture. A class of graphs is monadically stable if and only if it is a first-order transduction of a nowhere dense class of graphs.

Nešetřil–Ossona de Mendez's strong mass-transport conjecture. Every first-order convergent sequence of graphs from ${\cal C}$ has a modeling limit that satisfies the strong finitar…

Let $\mathcal{C}$ be a nowhere dense graph class, and let $\mathcal{D}$ be a graph class FO interpretable in $\mathcal{C}$. FO model-checking conjecture. The class $\mathcal{D}$ ha…

The neighborhood-complexity characterization. The class $\mathcal{C}$ is nowhere dense if and only if there exists a function $f_{\mathrm{nei}}(r,\varepsilon)$ such that

The nowhere-dense domination-set kernelization dichotomy. If $\mathcal{C}$ is nowhere dense, then for each $r\in\mathbb{N}$, the textsc{Distance-$r$ Dominating Set} problem admits…

Strong mass-transport conjecture. One can require a version of the strong mass transport principle. The text explains that the inequality is immediate for first-order definable set…

Let $(G_n)_{n\in\mathbb{N}}$ be an FO-convergent sequence of graphs, and suppose that the class $\{G_n:n\in\mathbb{N}\}$ is nowhere dense. Nowhere-dense modeling FO-limit conjectur…