@Kratochvil91a] as the intersection graphs of a set of curves in the plane that lie inside a disk such that each curve intersects the boundary of the disk in one of its endpoints. Keil et al. [@KeilMPV15] gave an $O(N^3)$-time algorithm for [<span style="font-variant:small-caps;">Maximum Independent Set</span> ]{}on outerstring graphs, where they model the disk as a polygon and each curve as a polygonal line (i.e., a chain of line segments) completely inside the polygon such that one of its endpoints coincides with a vertex of the polygon. Here, $N$ denotes the total number of segments used in the geometric representation of the polygonal lines and the polygon. The main part of the proof of Lemma \[lem:misOnIM\] is to show that $G_{I_M}$ is an outerstring graph.

\[lem:misOnIM\] [<span style="font-variant:small-caps;">Maximum Independent Set</span> ]{}can be solved on $G_{I_M}$ in $O(N^3)$ time, where $N$ is the number of paths in $I_M$.

We prove that $G_{I_M}$ is an outerstring graph with an $O(N)$ geometric representation. The lemma then follows by the algorithm of Keil et al. [@KeilMPV15] for solving [<span style="font-variant:small-caps;">Maximum Independent Set</span> ]{}on outerstring graphs.

Enclose the paths in $I_M$ within a large-enough vertical rectangle $R$ such that the vertical line $\ell$ contains the 