  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 representations behave very differently.
  The bounded representation problem belongs to a wider class of restricted
representation problems. These problems are generalizations of the
well-understood recognition problem, and they ask whether there exists a
representation of G satisfying some additional constraints. The bounded
representation problems generalize many of these problems.
