# Multitolerance Graphs

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2011; Mertzios

## Problem Definition

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. A graph G = (V, E) on n vertices is a tolerance graph if there exists a collection I = { I v  | vV } of closed intervals on the real line and a set t = { t v  | vV } of positive numbers, such that for any two vertices u, vV , uvE if and only if $$\vert I_{u} \cap I_{v}\vert \geq \min \{ t_{u},t_{v}\}$$, where | I | denotes the length of the interval I.

Tolerance graphs have been introduced in [3], in order to generalize some of the well-known applications of interval graphs. If in the definition of tolerance graphs we replace the operation “min” between tolerances by “max,” we obtain the class of max-tolerance graphs [7]. Both tolerance and max-tolerance graphs have attracted many research efforts (e.g., [4, 5, 710]) as they find numerous applications,...

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4. Golumbic MC, Siani A (2002) Coloring algorithms for tolerance graphs: reasoning and scheduling with interval constraints. In: Proceedings of the joint international conferences on artificial intelligence, automated reasoning, and symbolic computation (AISC/Calculemus), Marseille, pp 196–207

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8. Lehmann KA, Kaufmann M, Steigele S, Nieselt K (2006) On the maximal cliques in c-max-tolerance graphs and their application in clustering molecular sequences. Algorithms Mol Biol 1:9

9. Mertzios GB, Sau I, Zaks S (2009) A new intersection model and improved algorithms for tolerance graphs. SIAM J Discret Math 23(4):1800–1813

10. Mertzios GB, Sau I, Zaks S (2011) The recognition of tolerance and bounded tolerance graphs. SIAM J Comput 40(5):1234–1257

11. Parra A (1998) Triangulating multitolerance graphs. Discret Appl Math 84(1–3):183–197

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Correspondence to George B. Mertzios .

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Mertzios, G.B. (2014). Multitolerance Graphs. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-3-642-27848-8_684-1

• DOI: https://doi.org/10.1007/978-3-642-27848-8_684-1

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• Publisher Name: Springer, Boston, MA

• Online ISBN: 978-3-642-27848-8

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