Simultaneous Representation of Proper and Unit Interval Graphs

TL;DR

This paper presents linear-time algorithms for recognizing simultaneous proper interval graphs and polynomial-time algorithms for simultaneous unit interval graphs in the sunflower case, addressing open complexity questions.

Contribution

It provides the first efficient recognition algorithms for simultaneous proper and unit interval graphs in the sunflower case, and establishes NP-completeness in the general case.

Findings

Recognition of simultaneous proper interval graphs is linear-time in the sunflower case.

Recognition of simultaneous unit interval graphs is polynomial-time in the sunflower case.

Both problems are NP-complete when the number of graphs is not fixed.

Abstract

In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where all objects share the same common structure. While the recognition problem for general simultaneous interval graphs -- the simultaneous version of arguably one of the most well-studied graph classes -- is NP-complete, the complexity of the sunflower case for three or more simultaneous interval graphs is currently open. In this work we settle this question for proper interval graphs. We give an algorithm to recognize simultaneous proper interval graphs in linear time in the sunflower case where we allow any number of simultaneous graphs. Simultaneous unit interval graphs are much more 'rigid' and therefore have less freedom in their representation. We show they can be recognized in time O(|V|*|E|) for any number of simultaneous graphs in the sunflower case where G = (V, E) is the union of the simultaneous graphs. We further show that both recognition problems are in general NP-complete if the number of simultaneous graphs is not fixed. The restriction to the sunflower case is in this sense necessary.