On local structures of cubicity 2 graphs

TL;DR

This paper investigates the local structures of cubicity 2 graphs, specifically focusing on 2-stab unit interval graphs, and provides a polynomial time recognition algorithm for trees with such representations.

Contribution

It introduces the concept of 2-stab unit interval graphs and offers the first polynomial time algorithm for recognizing trees with cubicity 2.

Findings

Recognition of trees with 2SUIG representation is polynomial time.

The complexity of recognizing cubicity 2 in general trees remains open.

2SUIG graphs help understand local structures of cubicity 2 graphs.

Abstract

A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the $X$-axis, distance $1 + \epsilon$ ($0 < \epsilon < 1$) apart. This family of graphs allow us to study local structures of unit square intersection graphs, that is, graphs with cubicity 2. The complexity of determining whether a tree has cubicity 2 is unknown while the graph recognition problem for unit square intersection graph is known to be NP-hard. We present a polynomial time algorithm for recognizing trees that admit a 2SUIG representation.