of elements of $\mathcal{U}$ hitting a cross $c$ and let $\mathcal{S}$ be the collection of these sets; notice that there is a set in $\mathcal{S}$ for each cross $c$. This forms the set system $(\mathcal{U}, \mathcal{S})$ corresponding to [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}on $G$. For a set $S\in\mathcal{S}$, we say that an element $e\in\mathcal{U}$ *intersects* $S$ if $e\in S$.

\[lem:ifMDSthenMHS\] If there is a dominating set of size $k$ on the one-string [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graph $G$, then there is a hitting set of size at most $2k$ on the set system $(\mathcal{U}, \mathcal{S})$.

\[prf:lem:ifMDSthenMHS\] Let $M$ be a dominating set of size $k$ for $G$. For each path $P\in M$, add the supporting segments of ${\texttt{cross}}(P)$ into $M'$. Clearly, $\lvert M'\rvert=2k$. Now, consider a set $S\in\mathcal{S}$ and let $P$ and $c$ denote its corresponding path and cross, respectively. Since $M$ is a dominating set, either $P\in M$ or $P'\in M$ for some path $P'$ intersecting $P$. If $P\in M$, then clearly $S$ is intersected by some segment in $M'$. If $P'\in M$, then $c$ intersects ${\texttt{cross}}(P')$ by Observation \[obs:crossAndPath\] and so at least one of the supporting segments of ${\texttt{cross}}(P')$ (that is in $M'$) intersects $S$.

\[lem:ifMHSthenMDS\] If there is 