Computing maximum independent set on outerstring graphs and their relatives

A graph G with n vertices is called an outerstring if it has intersection representation a set of curves inside disk such that one endpoint every curve attached to the boundary disk. Given s segments, Maximum Independent Set ( MIS ) can be computed in O 3 time (Keil et al. (2017) [22] ). We examine fine-grained complexity problem on some well-known representations (e.g., line L -shapes, etc.), where strings are constant size. show computing grounded segment and square- at least as hard circle representations. Note no 2 − δ -time algorithm, > 0 , known for graphs. For string representations, y -monotone simple polygonal paths length segments integral coordinates, we solve this best possible under Strong Exponential Time Hypothesis. -shapes plane, give 4 ⋅ log ⁡ OPT -approximation algorithm (where denotes size optimal solution), improving previously best-known Biedl Derka (WADS 2017).