A d-box is the Cartesian product of d intervals of \(\mathbb {R}\) and a d-box representation of a graph G is a representation of G as the intersection graph of a set of d-boxes in \(\mathbb {R}^d\). It was proved by Thomassen in 1986 that every planar graph has a 3-box representation. In this paper we prove that every graph embedded in a fixed orientable surface, without short non-contractible cycles, has a 5-box representation. This directly implies that there is a function f, such that in every graph of genus g, a set of at most f(g) vertices can be removed so that the resulting graph has a 5-box representation. We show that such a function f can be made linear in g. Finally, we prove that for any proper minor-closed class \(\mathcal {F}\), there is a constant \(c(\mathcal {F})\) such that every graph of \(\mathcal {F}\) without cycles of length less than \(c(\mathcal {F})\) has a 3-box representation, which is best possible.
['Louis Esperet']
Box representations of embedded graphs