Dice Question Streamline Icon: https://streamlinehq.com

FPT status of shortest genus‑splitting curve on a surface

Establish whether computing a shortest weakly simple closed curve on an orientable combinatorial surface of genus g that cuts off a subsurface of genus g′ (for 1 ≤ g′ < g) is fixed‑parameter tractable with respect to the parameter g′.

Information Square Streamline Icon: https://streamlinehq.com

Background

This question concerns the parameterized complexity of finding a shortest weakly simple closed curve that separates a subsurface of prescribed genus on an orientable combinatorial surface. Prior work shows the problem is fixed‑parameter tractable (FPT) when parameterized by the total genus g of the surface.

The present paper’s Graph‑Enclosure‑with‑Penalties algorithm yields FPT results for certain special instances related to this surface‑splitting task, giving evidence toward tractability in g′ but not resolving the general case, which the authors explicitly note remains open.

References

Our result may shed some light on the following open problem by Bulavka, Colin de Verdière, and FuladiConclusion: given an orientable combinatorial surface of genus $g$, and an integer~$g'$, $1\le g'<g$, is it FPT in~$g'$ to compute a shortest weakly simple closed curve that cuts off a surface of genus~$g'$? ... and thus provides some hope for a positive answer in general, although this remains open.

— Finding a Shortest Curve that Separates Few Objects from Many (2504.03558 - Biedl et al., 4 Apr 2025) in Section 9, Applications, Extensions, and Open Problems (Splitting a surface)