Multitolerance Graphs

Years and Authors of Summarized Original Work

• 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 $I =\{ I_{v}\ \vert \ v \in V \}$ of closed intervals on the real line and a set $t =\{ t_{v}\ \vert \ v \in V \}$ of positive numbers, such that for any two vertices u, v ∈ V , u v ∈ E 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, 7–10]) as they...