$P_u, P_v\in P$ share a grid edge of $\mathcal{G}$ if and only if $(u, v)\in E$.

Instead of considering the edge intersection of graphs on a grid, we can think of the vertex intersection of graphs on a grid. More formally, a graph is said to have a VPG representation (stands for Vertex representation of Paths in a Grid), if its vertices can be represented as simple orthogonal paths on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one grid node. Although these graphs were considered a while ago when studying string graphs [@MiddendorfP92], they were formally investigated by Asinowski et al. [@AsinowskiCGLLS12]. Similar to B$_k$-EPG graphs, a *B$_k$-VPG graph* is a VPG graph in which each path has at most $k$ bends. In this paper, we are interested in B$_k$-VPG graphs for $k=1$.

\[def:B1VPG\] A graph $G=(V, E)$ is called a *[<span style="font-variant:small-caps;">B$_1$-VPG</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

two paths $P_u$ and $P_v$ intersect each other at a grid node if and only 