Simultaneous proper interval graphs can be recognized in linear time and simultaneous unit interval graphs in O(|V||E|) time in the sunflower case, and both become NP-complete without the sunflower restriction.
A note on simultaneous representation problem for interval and circular-arc graphs
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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.
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Simultaneous Representation of Proper and Unit Interval Graphs
Simultaneous proper interval graphs can be recognized in linear time and simultaneous unit interval graphs in O(|V||E|) time in the sunflower case, and both become NP-complete without the sunflower restriction.