General results on tolerance intersection graphs

In this paper, general results are presented that are related to ϕ-tolerance intersection graphs previously defined by Jacobson, McMorris, and Mulder. For example, it is shown that all graphs are ϕ-tolerance intersection graphs for all ϕ, yet for “nice” ϕ, almost no graphs are ϕ-tolerance interval graphs. Additional results about representation of trees are given.