Efficient solvability of MIN-MEG on broader graph classes

Determine whether the Minimum Monitoring Edge-Geodetic Set problem is solvable in polynomial time for circle graphs, permutation graphs, chordal graphs, cocomparability graphs, and the other graph classes identified in the paper as having open complexity status.

Background

The paper develops polynomial-time algorithms for MIN-MEG on distance-hereditary graphs, P4-sparse graphs, bipartite permutation graphs, and strongly chordal graphs. It explicitly identifies several broader or related graph classes—including circle graphs, permutation graphs, chordal graphs, and cocomparability graphs—for which the computational complexity of MIN-MEG remains unresolved. The open problem asks whether efficient algorithms exist for these classes.

References

As can be seen in Figure~\ref{fig:diagram}, the complexity status of the MIN-MEG problem is open for graph classes such as circle graphs, permutation graphs, chordal graphs, cocomparability graphs, etc. Is the MIN-MEG problem efficiently solvable for these graph classes?

— Characterizing optimal monitoring edge-geodetic sets for some structured graph classes  (2503.06086 - Foucaud et al., 8 Mar 2025) in Question 3, Section 6, “Conclusion and future aspects”