Abstract
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 intervals by lower and upper bounds. We show that this problem is -complete for disconnected input graphs and give a polynomial-time algorithm for a special class of instances, where the ordering of the connected components of the input graph along the real line is fixed.
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 , proper and unit interval representations have very different structure.
This explains why partial representation extension problems for these different types of representations require substantially different techniques.