Abstract
Tolerance graphs model interval relations in such a way that intervals can tolerate a certain amount of overlap without being in conflict.
In one of the most natural generalizations of tolerance graphs with direct applications in the comparison of DNA sequences from different organisms, namely multitolerance graphs, two tolerances are allowed
for each interval  one from the left and one from the right side.
Several efficient algorithms for optimization problems that are NPhard in general graphs have been designed for tolerance and multitolerance graphs. In spite of this progress, the complexity status of some fundamental algorithmic problems on tolerance and multitolerance graphs, such as the dominating set problem, remained unresolved until now,
three decades after the introduction of tolerance graphs. In this article we introduce two new geometric representations for tolerance and multitolerance graphs, given by points and line segments in the plane.
Apart from being important on their own, these new representations prove to be a powerful tool for deriving both hardness results and polynomial time algorithms. Using them, we surprisingly prove that the dominating set problem can be solved in polynomial time on tolerance graphs and that it is APXhard on multitolerance graphs, solving thus a longstanding open problem. This problem is the first one that has been discovered with a different complexity status in these two graph classes. Furthermore we present an algorithm that solves the independent dominating set problem on multitolerance graphs in polynomial time,
thus demonstrating the potential of this new representation for further exploitation via sweep line algorithms.
BibTeX  Entry
@InProceedings{giannopoulou_et_al:LIPIcs:2015:4926,
author = {Archontia C. Giannopoulou and George B. Mertzios},
title = {{New Geometric Representations and Domination Problems on Tolerance and Multitolerance Graphs}},
booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
pages = {354366},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897781},
ISSN = {18688969},
year = {2015},
volume = {30},
editor = {Ernst W. Mayr and Nicolas Ollinger},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/4926},
URN = {urn:nbn:de:0030drops49268},
doi = {10.4230/LIPIcs.STACS.2015.354},
annote = {Keywords: tolerance graphs, multitolerance graphs, geometric representation, dominating set problem, polynomial time algorithm, APXhard}
}
Keywords: 

tolerance graphs, multitolerance graphs, geometric representation, dominating set problem, polynomial time algorithm, APXhard 
Seminar: 

32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015) 
Issue Date: 

2015 
Date of publication: 

25.02.2015 