The Clique Problem in Ray Intersection Graphs

Ray intersection graphs are intersection graphs of rays, or halflines, in the plane. We show that any planar graph has an even subdivision whose complement is a ray intersection graph. The construction can be done in polynomial time and implies that finding a maximum clique in a segment intersection graph is NP-hard. This solves a 21-year old open problem posed by Kratochv'il and Ne\v{s}et\v{r}il.

💡 Research Summary

The paper resolves a long‑standing open problem concerning the computational complexity of the maximum clique problem in geometric intersection graphs. While the maximum independent set problem is known to be NP‑hard even for segment intersection graphs, the status of the maximum clique problem for general segment (or ray) intersection graphs remained unresolved for over two decades. The authors prove that the problem is NP‑hard even when the underlying geometric objects are rays (half‑lines), and consequently also for segment intersection graphs.

The core of the proof is a structural reduction: every planar graph G can be transformed, via an even subdivision (each edge is subdivided an even number of times), into a graph G′ whose complement (\overline{G′}) is a ray‑intersection graph. Because the maximum independent set problem is NP‑hard on planar graphs, this reduction yields a polynomial‑time many‑one reduction from planar‑graph independent set to the maximum clique problem on ray‑intersection graphs.

The construction proceeds in several stages:

1. Reference Frame – Choose an odd integer k and place k equally spaced points on the unit circle. For each i, define a rectangle (R_i) and a circular arc (\alpha_i) that is tangent to both diagonals of (R_i). The family of arcs ({\alpha_i}) together with the sets (\Gamma_i) of rays that intersect (\alpha_i) twice (or are tangent) and whose origins lie on (\alpha_{i+1}) form a “reference frame”. Lemmas show that any ray from (\Gamma_i) intersects any ray from (\Gamma_j) when (|i-j|>1), and that tangency properties give precise control over intersections.

2. Snooker Representation of Trees – For an embedded tree T equipped with a natural linear order τ (derived from a rooted BFS traversal and a rotation system), each vertex at level i is represented by a ray from (\Gamma_i). The origin of the ray lies on (\alpha_{i+1}); a child’s ray passes through its parent’s origin, guaranteeing that parent‑child pairs do not intersect (the rays are open). Vertices on the same level have origins that are consecutive along (\alpha_{i+1}); this ordering forces rays with different parents to intersect, while those sharing a parent intersect only at the parent’s origin. Lemma 2.3 proves that the complement of any embedded tree admits such a representation.

3. Admissible Extensions – The tree representation is extended by adding a set P of short paths (each of length 3 or 4). Each path connects two consecutive leaf vertices of T that lie on the same level. The internal vertices of a path are not part of T. The authors show how to realize each such path with one or two additional rays that intersect all other rays except the two leaf rays they connect, preserving the “snooker” properties. Lemma 2.4 establishes that the complement of (T+P) is a ray‑intersection graph.

4. Even Subdivision of an Arbitrary Planar Graph – Given any planar graph G, a BFS tree B is taken from an arbitrary outer‑face vertex. The remaining edges C form a spanning tree in the dual graph. Each edge in C is subdivided an even number of times, determined recursively from the dual tree depth (4 for leaf edges, otherwise 2 plus the maximum of its descendants). The resulting graph H is an even subdivision of G that can be decomposed into a tree T (the BFS tree) and an admissible extension P (the subdivided non‑tree edges). Lemma 2.5 shows that this decomposition can be computed in polynomial time.

Putting the pieces together, the complement of H is a ray‑intersection graph, and any maximum independent set in G corresponds to a maximum clique in that ray graph. Since finding a maximum independent set in planar graphs is NP‑hard, the maximum clique problem for ray (and therefore segment) intersection graphs is NP‑hard.

The paper also discusses related work, noting that while complements of planar graphs are known to be intersection graphs of convex polygons, it remains open whether every planar graph’s complement can be realized as a segment intersection graph. The authors’ construction provides a new structural insight that may be useful for studying other geometric intersection graph classes.

In summary, the authors deliver a polynomial‑time reduction from planar‑graph independent set to ray‑intersection graph clique, thereby proving NP‑hardness of the maximum clique problem for ray and segment intersection graphs and finally settling the 21‑year‑old open problem posed by Kratochvíl and Nešetřil.