Simultaneous Representation of Proper and Unit Interval Graphs

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
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graphs is not fixed. The restriction to the sunflower case is in this sense necessary