Linear-time Algorithm for Partial Representation Extension of Interval Graphs

Interval graphs are intersection graphs of closed intervals of the real-line. The wellknown computational problem, called recognition, asks for an input graph G whether it can be represented by closed intervals, i.e., whether G is an interval graph. There are several linear-time algorithms known for recognizing interval graphs. In this paper, we study a generalization of recognition, called partial representation extension. Input of this problem consists of a graph G with a partial representation R fixing positions of some intervals. The problem asks whether it is possible to place the remaining interval and create an interval representation R of the entire graph G extending R. We give a linear-time algorithm based on PQ-trees which solves this problem.