On edge intersection graphs of paths with 2 bends

TL;DR

This paper investigates the complexity of recognizing edge intersection graphs of paths with at most 2 bends in a grid, introduces unaligned polyline representations, and explores the trade-offs between bends and slopes.

Contribution

It establishes the hardness of recognition for EPG graphs with 2 bends and studies new classes of polyline representations with slope constraints.

Findings

Recognition of 2-bend EPG graphs is NP-hard.

Recognition remains hard for subclasses of these graphs.

Trade-offs exist between the number of bends and slopes in polyline representations.

Abstract

An EPG-representation of a graph $G$ is a collection of paths in a plane square grid, each corresponding to a single vertex of $G$, so that two vertices are adjacent if and only if their corresponding paths share infinitely many points. In this paper we focus on graphs admitting EPG-representations by paths with at most 2 bends. We show hardness of the recognition problem for this class of graphs, along with some subclasses. We also initiate the study of graphs representable by unaligned polylines, and by polylines, whose every segment is parallel to one of prescribed slopes. We show hardness of recognition and explore the trade-off between the number of bends and the number of slopes.