  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.
