Unsolved Problems in Visibility Graphs of Points, Segments and Polygons

In this survey paper, we present open problems and conjectures on visibility graphs of points, segments and polygons along with necessary backgrounds for understanding them.

💡 Research Summary

The surveyed paper provides a comprehensive catalogue of open problems and conjectures concerning visibility graphs (VGs) of points, line segments, and polygons, together with the necessary background to understand each issue. It begins by recalling the definition of a visibility graph: two geometric objects are adjacent if the straight‑line segment joining them lies entirely within a prescribed region (the whole plane, a polygon interior, etc.). The authors stress the relevance of VGs to fields such as computer graphics, robot motion planning, and wireless network design, and they summarize the basic structural properties that have already been established—symmetry, triangle inequality, planarity in certain cases, and known bounds on degree, chromatic number, and edge density.

The first major section deals with point visibility graphs (PVGs). Three families of questions dominate this part: (1) the recognition problem—given an undirected graph G, does there exist a set S of points in the plane whose visibility graph is exactly G?—for which no general complexity classification is known; only special graph classes (trees, cycles, bipartite graphs) admit polynomial‑time algorithms. (2) The reconstruction problem, which asks for an explicit point placement realizing a given PVG; current approaches are largely heuristic or fixed‑parameter, and exact algorithms are missing. (3) Structural bounds such as maximum clique size, minimum chromatic number, and degree constraints. While several asymptotic inequalities have been proved, tight constants remain unknown, and many conjectures (e.g., “every PVG with n vertices has a clique of size at least √n”) are still open.

The second section focuses on segment visibility graphs (SVGs), distinguishing the non‑crossing case from the crossing‑allowed case. In the non‑crossing setting the resulting graphs are always planar, but once segment intersections are permitted the class expands dramatically, and the authors highlight two central open problems: (i) “Are all SVGs planar?”—no counter‑example has been found, yet the conjecture lacks proof; (ii) “What is the computational complexity of recognizing SVGs?”—the best known upper bound places the problem in PSPACE, and a PSPACE‑hardness reduction has been sketched, suggesting the problem is unlikely to be in NP. Additional questions concern the effect of length or orientation restrictions on degree and chromatic number, for which only partial results exist.

The third and most intricate part concerns polygon visibility graphs (PVGs), both interior and exterior visibility. For convex polygons the visibility graph is complete, but as soon as holes or reflex vertices appear the structure becomes highly non‑trivial. The paper lists four flagship conjectures: (1) Recognition Complexity – is the problem NP‑complete for general simple polygons? (2) Recognition Algorithms for Restricted Families – e.g., simple polygons with a single hole admit an O(n³) algorithm, but extending this to multiple holes remains open. (3) Four‑Color Conjecture – every polygon visibility graph is conjectured to be 4‑colorable; proved for convex, star‑shaped, and some single‑hole polygons, but unproved in general. (4) Hamiltonicity – no counter‑example is known, leading to the bold conjecture that every polygon visibility graph contains a Hamiltonian cycle. The authors also discuss known lower and upper bounds on edge density, degree sequences, and the relationship between polygon geometry (e.g., number of reflex vertices) and graph parameters.

In the final section the authors synthesize methodological trends and propose future research directions. They emphasize the role of parameterized algorithms (parameterizing by number of holes, reflex vertices, or visibility degree), embedding techniques (stretchability, planar embeddings), and randomized/approximation methods for reconstruction. A particularly promising line is the introduction of visibility matrices and distance‑visibility graphs, which simultaneously encode adjacency and Euclidean distance information; early results suggest these tools may break new ground on recognition and reconstruction problems. The paper calls for (a) a systematic study of mixed models (e.g., point‑segment‑polygon hybrids), (b) extensions to three‑dimensional visibility graphs, and (c) tighter integration of theoretical insights with practical applications such as optimal sensor placement and autonomous navigation. By cataloguing the open landscape and highlighting recent methodological advances, the survey sets a clear agenda for researchers aiming to resolve the deep combinatorial and algorithmic challenges that still dominate the theory of visibility graphs.