by Chalopin and Goncalves [@ChalopinG09].

EPG graphs (stands for Edge intersection of Paths in a Grid) were introduced by Golumbic et al. [@GolumbicLS09] as the class of graphs whose vertices can be represented as simple orthogonal paths on a rectangular grid in such a way that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. Golumbic et al. [@GolumbicLS09] showed that every graph is an EPG graph, and the size of the underlying grid is polynomial in the size of $G$. Since then much of the work has focused on studying subclasses of EPG graphs; in particular, restricting the type of paths allowed. A turn of a path at a grid node is called a *bend* and a graph is called a *B$_k$-EPG graph* if it has an EPG representation in which each path has at most $k$ bends. In this paper, we are interested in B$_k$-EPG graphs for $k=1$.

\[def:B1EPG\] A graph $G=(V,E)$ is called a *[<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graph*, if every vertex $u$ of $G$ can be represented as a *path* $P_u$ on a grid $\mathcal{G}$ such that

$P_u$ is orthogonal and has at most one bend, and

paths 