Bounded Representations of Interval and Proper Interval Graphs
Martin Balko, Pavel Klav\'ik, Yota Otachi

TL;DR
This paper investigates the bounded representation problem for interval and proper interval graphs, providing efficient algorithms for some cases and highlighting complexity differences between related graph classes.
Contribution
It introduces linear and quadratic time algorithms for bounded representations and reveals surprising complexity distinctions between proper and unit interval graphs.
Findings
Linear time algorithm for interval graphs
Quadratic time algorithm for proper interval graphs
NP-completeness for bounded representations of unit interval graphs
Abstract
Klavik et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals L_v and R_v called bounds. We ask whether there exists a bounded representation in which each interval I_v has its left endpoint in L_v and its right endpoint in R_v. We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs. Robert's Theorem states that the classes of proper interval graphs and unit interval graphs are equal. Surprisingly the bounded representation problem is polynomially solvable for proper interval graphs and NP-complete for unit interval graphs [Klav\'{\i}k et al., arxiv:1207.6960]. So unless P = NP, the proper and unit interval…
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