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

TL;DR

This paper investigates the properties of intersection graphs formed by k-bend paths on a grid, revealing their hierarchical structure, computational complexity, and relation to other graph classes.

Contribution

It establishes that B_k-VPG classes form a strict hierarchy, proves NP-completeness of recognition for these graphs, and clarifies their relation to segment intersection graphs.

Findings

B_k-VPG is a proper subset of B_{k+1}-VPG for fixed k

Recognition of B_k-VPG graphs is NP-complete

B_k-VPG classes are incomparable with segment intersection graphs

Abstract

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.