Living Reference Work Entry

Encyclopedia of Algorithms

pp 1-6

# Multitolerance Graphs

• George B. MertziosAffiliated withSchool of Engineering and Computing Sciences, Durham University Email author

### Keywords

Multitolerance graphs Tolerance graphs Intersection model Minimum coloring Maximum clique Maximum-weight independent set

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 = { Iv | vV } of closed intervals on the real line and a set t = { tv | 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, especially i ...

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