the circle and two vertices are adjacent if and only if their corresponding chords intersect. The proof of Asinowski et al. [@AsinowskiCGLLS12] shows that every circle graph is in fact a one-string [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graph (although not every one-string [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graph is a circle graph [@Bouchet94]). Since [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}is [<span style="font-variant:small-caps;">APX</span>-hard ]{}on circle graphs [@DamianP06], the problem becomes [<span style="font-variant:small-caps;">APX</span>-hard ]{}also on one-string [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graphs.[^1] However, to our knowledge, there is no approximation algorithm known for the problem. We note that there are $O(1)$-approximation algorithms for [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}on circle graphs [@Damian-IordacheP99a; @Damian-IordacheP02], but these algorithms do not work for [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graphs as they heavily rely on the fact that the vertices of the input graph are modelled as chords of a circle.

-   We give a polynomial-time $O(1)$-approximation algorithm for the [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}problem on *one-string* [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graphs. To our knowledge, this is the first constant-factor approximation algorithm for this problem on [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graphs.

Our algorithm is based on formulating the problem as a hitting set problem among orthogonal line segments and then proving that the corresponding hitting set has a small $\varepsilon$-net. However, proving the existence 