  The recently introduced problem of extending partial interval representations
asks, for an interval graph with some intervals pre-drawn by the input, whether
the partial representation can be extended to a representation of the entire
graph. In this paper, we give a linear-time algorithm for extending proper
interval representations and an almost quadratic-time algorithm for extending
unit interval representations.
  We also introduce the more general problem of bounded representations of unit
interval graphs, where the input constrains the positions of some intervals by
lower and upper bounds. We show that this problem is NP-complete for
disconnected input graphs and give a polynomial-time algorithm for the special
class of instances, where the ordering of the connected components of the input
graph along the real line is prescribed. This includes the case of partial
representation extension.
  The hardness result sharply contrasts the recent polynomial-time algorithm
for bounded representations of proper interval graphs [Balko et al. ISAAC'13].
So unless $\text{P} = \text{NP}$, proper and unit interval representations have
vastly different structure. This explains why partial representation extension
problems for these different types of representations require substantially
different techniques.
