Recognition of Unit Segment and Polyline Graphs is $\exists\mathbb{R}$-Complete

TL;DR

This paper proves that recognizing intersection graphs of unit segments and polylines with a fixed number of bends is computationally very hard, specifically $orall ext{R}$-complete, filling a gap in understanding the complexity of geometric graph recognition.

Contribution

It establishes the $orall ext{R}$-completeness of recognition problems for unit segments and polylines with fixed bends, a previously unresolved case in geometric intersection graph recognition.

Findings

Recognition problems are $orall ext{R}$-complete for these objects.

This completes the classification of recognition complexity for common geometric intersection graphs.

The result highlights the computational difficulty of geometric graph recognition tasks.

Abstract

Given a set of objects $O$ in the plane, the corresponding intersection graph is defined as follows. Each object defines a vertex and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit segments and polylines with exactly $k$ bends. In the recognition problem, we are given a graph and want to decide whether the graph can be represented as an intersection graph of certain geometric objects. In previous work it was shown that various recognition problems are $\exists\mathbb{R}$-complete, leaving unit segments and polylines among the few remaining natural cases where the recognition complexity remained open. We show that recognition for both families of objects is $\exists\mathbb{R}$-complete.