  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.
