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.
On some NP -complete SEFE problems
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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.