A note on simultaneous representation problem for interval and circular-arc graphs

TL;DR

This paper proves that the problem of simultaneously representing multiple interval and circular-arc graphs with shared vertices is NP-complete when the number of graphs is part of the input and the graphs are not in sunflower position.

Contribution

It establishes NP-completeness results for the simultaneous representation problem for interval and circular-arc graphs under specific conditions.

Findings

NP-completeness for simultaneous representation of interval graphs

NP-completeness for simultaneous representation of circular-arc graphs

Results apply when the number of graphs is variable and graphs are not in sunflower position

Abstract

In this short note, we show two NP-completeness results regarding the \emph{simultaneous representation problem}, introduced by Lubiw and Jampani. The simultaneous representation problem for a given class of intersection graphs asks if some $k$ graphs can be represented so that every vertex is represented by the same interval in each representation. We prove that it is NP-complete to decide this for the class of interval and circular-arc graphs in the case when $k$ is a part of the input and graphs are not in a sunflower position.