Extending Partial Representations of Interval Graphs

TL;DR

This paper extends the recognition problem for interval graphs to partial representations, providing a linear-time algorithm for extending fixed partial interval placements to full representations.

Contribution

It generalizes the characterization of interval graphs to partial representations and introduces a linear-time algorithm for extension based on PQ-trees.

Findings

Linear-time algorithm for partial representation extension.

Generalization of interval graph characterization.

Extension of recognition algorithms to partial representations.

Abstract

Interval graphs are intersection graphs of closed intervals of the real-line. The well-known computational problem, called recognition, asks whether an input graph $G$ 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, the oldest one is by Booth and Lueker [J. Comput. System Sci., 13 (1976)] based on PQ-trees. In this paper, we study a generalization of recognition, called partial representation extension. The input of this problem consists of a graph $G$ with a partial representation $\cal R'$ fixing the positions of some intervals. The problem asks whether it is possible to place the remaining interval and create an interval representation $\cal R$ of the entire graph $G$ extending $\cal R'$. We generalize the characterization of interval graphs by Fulkerson and Gross [Pac. J. Math., 15 (1965)] to extendible partial representations. Using it, we give a linear-time algorithm for partial representation extension based on a reordering problem of PQ-trees.