Completion of the mixed unit interval graphs hierarchy

TL;DR

This paper introduces a new class within the hierarchy of mixed unit interval graphs, providing a complete characterization and quadratic-time recognition algorithms, advancing understanding of interval graph classes.

Contribution

It identifies and characterizes the missing class in the mixed unit interval graphs hierarchy and offers efficient recognition algorithms.

Findings

Complete characterization of the new class

Quadratic-time recognition algorithms

Extension of previous work to recognize mixed unit interval graphs

Abstract

We describe the missing class of the hierarchy of mixed unit interval graphs, generated by the intersection graphs of closed, open and one type of half-open intervals of the real line. This class lies strictly between unit interval graphs and mixed unit interval graphs. We give a complete characterization of this new class, as well as quadratic-time algorithms that recognize graphs from this class and produce a corresponding interval representation if one exists. We also mention that the work in arXiv:1405.4247 directly extends to provide a quadratic-time algorithm to recognize the class of mixed unit interval graphs.