Characterization of visibility intersection graphs

Determine whether every trapezoid graph is the visibility intersection graph of some weak visibility polygon, thereby characterizing exactly the conflict structure of the Discrete Witness Set Problem.

Background

The paper proves that the visibility intersection graph of any finite witness set in a weak visibility polygon is a trapezoid graph. It also proves that the class of such visibility intersection graphs properly contains both the interval graphs and the permutation graphs.

The remaining issue is whether the containment of visibility intersection graphs within trapezoid graphs is strict. Resolving this would determine whether the trapezoid representation captures exactly all possible conflict structures for the discrete witness set problem.

References

Whether the third containment in \cref{F-eq:vig-location} is proper is open. A positive answer would characterize the conflict structure of Discrete Witness Set exactly.

— Witness Set in Weak Visibility Polygons is Polynomial-Time Solvable  (2609.24460 - Das et al., 21 Sep 2026) in Remark following Theorem 4.12, Section 4.4 ("The Class of Visibility Intersection Graphs")