NDLI: New Geometric Representations and Domination Problems on Tolerance and Multitolerance Graphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Giannopoulou, Archontia C. Mertzios, George B.
Copyright Year	2016
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 on the left side and the other on the right side. Several efficient algorithms for optimization problems that are NP-hard 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 paper 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 APX-hard on multitolerance graphs, thus solving a longstanding open problem. This problem is the first one that has been discovered with a different complexity status in these two graph classes.
Sponsorship	Engineering and Physical Sciences Research Council
Starting Page	1685
Ending Page	1725
Page Count	41
File Format	PDF
ISSN	08954801
DOI	10.1137/15M1039468
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	3
Volume Number	30
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-08-30
Access Restriction	Subscribed
Subject Keyword	tolerance graph multitolerance graph geometric representation dominating set problem Graph theory APX-hard Graph algorithms Graph representations polynomial time algorithm
Content Type	Text
Resource Type	Article
Subject	Mathematics

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