Minimal Obstructions for Partial Representations of Interval Graphs

TL;DR

This paper characterizes minimal obstructions preventing partial interval graph representations from being extendible, generalizing classical forbidden subgraph characterizations and enabling efficient certifying algorithms.

Contribution

It introduces a complete characterization of minimal obstructions for partial interval graph representations, extending classical results and providing a linear-time certifying extension algorithm.

Findings

Minimal obstructions combine forbidden subgraphs with up to four pre-drawn intervals.

A Helly-type theorem states extendibility depends on quadruples of pre-drawn intervals.

The characterization enables a linear-time certifying extension algorithm.

Abstract

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension.