are [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}is also known by a complete family of minimal forbidden induced sub-graphs [@CohenGR14]. We observe by the construction of Ehrlich et al. [@EhrlichET76] that every planar graph has a VPG representation. Moreover, Chaplick and Ueckerdt [@ChaplickU13] proved that every planar graph has a B$_2$-VPG representation, although the representation may not be one-string. On the other hand, Chalopin et al. [@ChalopinGO10] showed that every planar graph has a one-string representation. Biedl and Derka [@BiedlD15] re-proved these results simultaneously by showing that every planar graph has a one-string B$_2$-VPG representation. It is however not known if every planar graph has also a one-string B$_1$-VPG representation. Felsner et al. [@FelsnerKMU16] proved that every planar 3-tree is [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}and the line graph of every planar graph is a [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graph.

In terms of recognition problems, it is known that recognizing [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs is <span style="font-variant:small-caps;">NP</span>-hard [@HeldtKU14], and the hardness of recognizing B$_2$-EPG graphs is shown by Pergel and Rz[ż]{}ewski [@PergelR16]. Moreover, recognizing VPG graphs is <span style="font-variant:small-caps;">NP</span>-complete [@ChaplickJKV12]. They also showed that recognizing if a given B$_{k+1}$-VPG graph is a B$_k$-VPG graph is <span style="font-variant:small-caps;">NP</span>-complete even if we are given a B$_{k+1}$-VPG representation as part of the input. 