Simultaneous Representation of Proper and Unit Interval Graphs

Abstract 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 $$\mathcal {O}(|V|\cdot |E|)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo>|</mml:mo> <mml:mi>V</mml:mi> <mml:mo>|</mml:mo> <mml:mo>·</mml:mo> <mml:mo>|</mml:mo> <mml:mi>E</mml:mi> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for any number of simultaneous graphs in the sunflower case where $$G=(V,E)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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.