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In Graph theory, a tolerance graph is a graph which is defined by a collection of intervals (vertices) and tolerances. The class of tolerance graphs is a generalization of the class of interval graphs, in the way that if the interval graphs are tolerance graphs with tolerances are constant. The tolerance graphs introduce the notion of tolerance in the conflicts, and thus add a new feature to the problems modelized by interval graphs. The tolerance graph are a study subject where there are still quite a lot of open questions, which make it an interesting subject, as there are a lot of potential discoveries.

Formal Definition
A graph $$G=(V,E)$$ is a tolerance graph if each vertex $$v \in V$$ can be assigned a closed interval $$I_v$$ and a tolerance $$t_v \in R^+$$ so that $$xy \in E$$ if and only if $$|I_x \cap I_y| \ge min \{t_x, t_y\}$$.

Such a collection $$  $$ of intervals and tolerances is called a tolerance representation where $$I=\{I_x | x \in V\}$$ and $$t=\{t_x | x\in V \}$$ [1]

Genesis of Tolerance Graphs
Tolerance graphs were introduced by Golumbic and Monma in 1982 to generalize some of the well-known applications associated with interval graphs. Their motivation introducing them was the need to solve scheduling problems in which resources such as rooms or vehicles may be needed on an exclusive basis, but where a measure of flexibility or tolerance would allow for sharing or relinquishing the ressource if a solution is not otherwise possible [1].

Tolerance representation of a tolerance graph
Despite the fact that we know that a tolerance graph has a tolerance representaton, it is yet unknown how to obtain this representation, given a graph known to be a tolerance graph. This has the consequence that sometimes the starting point is not a tolerance graph (with which we can't use a tolerance representation), but directly a tolerance representation, which we can use in algorithms.

Tolerance Graphs properties
Here are listed a few fundamental properties of tolerance graphs.

Hereditary Property
Being a tolerance graph is hereditary, i.e. any subgraph of a tolerance graph is a tolerance graph.

Forbidden Graphs

 * The cycle $$C_n$$ is not a tolerance graph for $$n \ge 5$$.
 * The graph $$ \overline{C_n}$$ is no a tolerance graph for $$n \ge 5$$.
 * The Berlin Graph $$ \overline{B} $$ is not a tolerance graph
 * $$ T_3 $$ is not a tolerance graph.
 * $$ \overline{T_3} $$ is not a tolerance graph.

Tolerance tree
If T is a tree and $$\overline{T}$$ is its complement, the following are equivalent:
 * 1) T is a tolerance graph.
 * 2) T has no subtree isomorphic to the graph $$T_3$$.
 * 3) $$\overline{T}$$ is a bounded tolerance graph.
 * 4) $$\overline{T}$$ is a tolerance graph.

Bipartite tolerance graph
Let G=(X,Y,E) be a bipartite graph. The following conditions are equivalent: The general question of characterizing bipartite tolerance graphs is still open.
 * 1) G is a bounded tolerance graph
 * 2) G is a trapezoid graph
 * 3) G is a cocomparability graph
 * 4) G is AT-free
 * 5) G is a permutation graph

Weakly chordal graphs
Tolerance graphs are weakly chordal.

Perfect Graphs
Tolerance graphs are perfect.

Co-Perfectly Orderable Graphs
Tolerance graphs are co-perfectly orderable.

Interval Graphs
Interval graphs are (bounded) tolerance graphs with constant tolerances.

Interval graphs are also unit tolerance graphs.

Permutation Graphs
Permutation graphs are tolerance graphs with $$ t_i = |I_i| $$ for all $$ i \in V(G) $$.

Parallelogram Graphs
Parallelogram graphs are exactly bounded tolerance graphs.

Interval Probe Graphs
An interval Probe graph is a tolerance graph (and an interval graph is an interval probe graph).

Recogition of Tolerance Graphs
Recognition of Tolerance graphs is in NP-complete.

Tolerance graph representation
It is not known how to obtain a tolerance representation when the input graph is known to be a tolerance graph

Optimal Coloring
We suppose that we have a tolerance representation of the graph. The algorithm is the following:


 * Input: a tolerance representation $$ < I, t > $$ for graph G=(V,E)


 * Output: a minimal coloring of the intervals

This algorithm is polynomial.
 * Method: Reduction and Coloring. In the reduction phase, when an interval of infinite tolerance is encountere, either its tolerance is lowered to the length of the interval or an arbitrary length. The coloring phase uses the algorithm of Felsner, Muller, and Wernisch(1997)

Maximum weight stable set
Given a tolerance representation of the graph, we can obtain a maximum weight stable set in polynomial time.

Particular graphs
The complete bipartite graph $$K_{3,3}$$ is a bounded tolerance graph, but not a proper tolerance graph.

$$T_2$$ is a tolerance graph

$$ \overline{T_2} $$ is a bounded tolerance graph

The cycle graph $$C_4$$ is a unit tolerance graph.

The complete bipartite graph $$ K_{1,3} $$ is a unit tolerance graph.

The path graph $$ P_4 $$ is a tolerance graph.

Applications
Tolerance graphs extend the notion of objects conflicting introduced by interval graphs by adding a 'tolerance' for the overlapping. Two objects conflict if and only if their area of conflict is larger than their area of tolerance. This introduces a notion of 'flexibility' in the applications used by interval graphs like applications datastorage, or genetics.

Open questions

 * Is there a cocomparability graph that is a tolerance graph but not a bounded tolerance graph ?
 * Characterization of bipartite tolerance graphs.
 * What is the complexity of the sandwich problem for tolerance graphs?
 * Characterization of complements of tolerance graphs.