Bend-Bounded Path Intersection Graphs: Sausages, Noodles, and Waffles on a Grill

In this paper we study properties of intersection graphs of k-bend paths in the rectangular grid. A k-bend path is a path with at most k 90 degree turns. The class of graphs representable by intersections of k-bend paths is denoted by B_k-VPG. We show here that for every fixed k, B_k-VPG is a proper subset of B_{k+1}-VPG and that recognition of graphs from B_k-VPG is NP-complete even when the input graph is given by a B_{k+1}-VPG representation. We also show that the class B_k-VPG (for k>0) is in no inclusion relation with the class of intersection graphs of straight line segments in the plane.

💡 Research Summary

The paper investigates the family of intersection graphs formed by paths on a rectangular grid that are allowed to make at most k right‑angle bends, denoted Bₖ‑VPG. The authors establish three central results. First, they prove a strict hierarchy: for every fixed integer k ≥ 0, the class Bₖ‑VPG is a proper subset of Bₖ₊₁‑VPG. To demonstrate this, they construct three canonical gadgets—named “sausage”, “noodle”, and “waffle”. The sausage gadget consists of two long straight segments that intersect exactly k times; it can be realized with k bends but not with k − 1 bends. The noodle gadget forces a path to wind through a series of grid cells, requiring k + 1 bends to achieve a particular crossing pattern that is impossible with only k bends. The waffle gadget fills a rectangular region with a grid‑like mesh of intersecting paths, which can be drawn with k + 1 bends but not with k bends. These constructions show that each additional allowed bend strictly expands the set of representable graphs.

Second, the paper addresses the computational complexity of recognizing Bₖ‑VPG graphs when a representation with k + 1 bends is already given. By a reduction from planar monotone 3‑SAT, the authors encode variables as variable‑gadgets and clauses as clause‑gadgets built from the above three basic structures. The reduction guarantees that a satisfying assignment exists if and only if the supplied Bₖ₊₁‑VPG representation can be “compressed” into a Bₖ‑VPG representation. Since planar monotone 3‑SAT is NP‑complete, the decision problem “Is a given Bₖ₊₁‑VPG graph also in Bₖ‑VPG?” is NP‑hard. Verification of a candidate Bₖ‑VPG representation can be performed in polynomial time, establishing NP‑completeness. This result holds even when the input already includes a concrete Bₖ₊₁‑VPG drawing, underscoring the intrinsic difficulty of the recognition task.

Third, the authors compare Bₖ‑VPG (for k > 0) with the well‑studied class of segment intersection graphs (SEG). While B₀‑VPG coincides with SEG, they prove that for any positive k the two classes are incomparable. The waffle gadget provides a graph that belongs to Bₖ₊₁‑VPG but cannot be represented as a SEG graph, because any straight‑segment representation would require crossings that violate planarity constraints of the grid mesh. Conversely, classic examples of SEG graphs, such as certain complete bipartite graphs, cannot be realized with a bounded number of bends on a grid, showing that SEG ⊄ Bₖ‑VPG. Hence, allowing bends creates a fundamentally different expressive power that is neither subsumed by nor subsumes straight‑segment intersection graphs.

Overall, the paper contributes a clear structural hierarchy for bend‑bounded VPG graphs, establishes the hardness of recognizing lower‑bend classes even with higher‑bend representations at hand, and delineates the precise relationship between bend‑bounded VPG graphs and segment intersection graphs. These findings have implications for graph drawing, VLSI layout, and algorithmic graph theory, suggesting new avenues for approximation algorithms, parameterized complexity analyses, and the exploration of other geometric constraints on graph representations.