1Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague; and Institute of Computer Science, The Czech Academy of Sciences, Czech Republic 2Department of Mathematics and European Centre of Excellence NTIS University of West Bohemia Pilsen Czech Republic.
The electronic journal of combinatorics, Tome 25 (2018) no. 4
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.
1
Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague; and Institute of Computer Science, The Czech Academy of Sciences, Czech Republic
2
Department of Mathematics and European Centre of Excellence NTIS
University of West Bohemia
Pilsen
Czech Republic.
@article{10_37236_5862,
author = {Pavel Klavik and Maria Saumell},
title = {Minimal obstructions for partial representations of interval graphs},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {4},
doi = {10.37236/5862},
zbl = {1409.05140},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5862/}
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Pavel Klavik; Maria Saumell. Minimal obstructions for partial representations of interval graphs. The electronic journal of combinatorics, Tome 25 (2018) no. 4. doi: 10.37236/5862
