Georgia Institute of Technology College of Sciences

Tolerance graphs were introduced in 1982 by Golumbic and Monma as a generalization of the class of interval graphs. A graph G= (V, E) is an interval graph if each vertex v \in V can be assigned a real interval I_v so that xy \in E(G) iff I_x \cap I_y \neq \emptyset. Interval graphs are important both because they arise in a variety of applications and also because some well-known recognition problems that are NP-complete for general graphs can be solved in polynomial time when restricted to the class of interval graphs. In certain applications it can be useful to allow a representation of a graph using real intervals in which there can be some overlap between the intervals assigned to non-adjacent vertices. This motivates the following definition: a graph G= (V, E) is a tolerance graph if each vertex v \in V can be assigned a real interval I_v and a positive tolerance t_v \in {\bf R} so that xy \in E(G) iff |I_x \cap I_y| \ge \min\{t_x,t_y\}. These topics can also be studied from the perspective of ordered sets, giving rise to the classes of Interval Orders and Tolerance Orders. In this talk we give an overview of work done in tolerance graphs and orders . We use hierarchy diagrams and geometric arguments as unifying themes.