Minimal Obstructions for Partial Representations of Interval Graphs
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.
Cite
@article{arxiv.1406.6228,
title = {Minimal Obstructions for Partial Representations of Interval Graphs},
author = {Pavel Klavík and Maria Saumell},
journal= {arXiv preprint arXiv:1406.6228},
year = {2016}
}