Unit interval graphs are those which have an interval representation in which every interval has the same length. Likewise, proper interval graphs are those which have an interval representation in which no interval properly contains another. Clearly the class of unit interval graphs is a subset of the class of proper interval graphs. In 1969, Fred Roberts showed that these classes are in fact equal.
We study tolerance and bitolerance graphs which generalize interval graphs by allowing a certain amount of overlap of intervals before an edge is formed between the corresponding vertices. We formalize this notion in the talk and discuss which classes have been characterized, which have efficient recognition algorithms, and then focus on the questions of when the unit and proper subclasses are equal.