MESP complexity on restricted planar graph classes

Determine the computational complexity of the Minimum Eccentricity Shortest Path problem on outerplanar graphs, series-parallel graphs, solid grids, and other important restricted classes of planar graphs.

Background

The paper establishes a polynomial-time algorithm for MESP on the class of K2,3K_{2,3}-minor-free graphs, which contains the outerplanar graphs. It contrasts this result with the known NP-completeness of MESP on subcubic partial grids and notes that the complexity on several other relevant planar classes has not been resolved.

This problem is included because determining whether MESP is tractable on these classes would clarify the boundary between the polynomial-time and NP-complete cases identified in the paper. The series-parallel case is particularly significant because it is equivalent to the class of graphs of treewidth at most two.

References

On the other hand, MESP remains NP-complete even on subcubic partial grids, a subclass of planar bipartite graphs of maximum degree at most $3$. However, the computational complexity of MESP on important restricted classes of planar graphs like outerplanar graphs, series-parallel graphs, solid grids etc., remain unknown.

— Minimum eccentricity shortest paths of $K_{2,3}$-minor-free graphs  (2608.13158 - Chakraborty et al., 13 Aug 2026) in Section 1, Introduction

Note that the computational complexity of MESP on graphs with constant treewidth is unknown. In fact, it is unknown if MESP admits a polynomial time algorithm on graphs of treewidth at most $2$, which are equivalent to series-parallel graphs.

— Minimum eccentricity shortest paths of $K_{2,3}$-minor-free graphs  (2608.13158 - Chakraborty et al., 13 Aug 2026) in Section 1, Introduction