Interval graphs have been studied extensively both because they have
many applications and because many problems which are NP-complete for
general graphs, admit polynomial-time solutions when restricted to the class
of interval graphs. Formally, a graph $G$ is an {\em interval
graph} if it can
be represented as follows: each vertex of $G$ corresponds to a real
interval so that two vertices are adjacent in $G$ if and only if their
corresponding intervals intersect.
{\em Unit interval graphs} are those which have an interval
representation in which every interval has the same length. Likewise,
{\em 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.