On Comparable Box Dimension

Two boxes in ${\mathbb{R}}^d$ are comparable if one of them is a subset of a translation of the other one. The comparable box dimension of a graph $G$ is the minimum integer $d$ such that $G$ can be represented as a touching graph of comparable axis-aligned boxes in ${\mathbb{R}}^d$. We show that proper minor-closed classes have bounded comparable box dimensions and explore further properties of this notion.