  Max-point-tolerance graphs (MPTG) was introduced by Catanzaro et al. in 2017
as a generalization of interval graphs. This graph class has many practical
applications in study of human genome as well as in signal processing for
networks. The same class of graphs were also studied in the name of p-BOX(1)
graphs by Soto and Caro in 2015. In our article, we consider a natural subclass
of max-point-tolerance graphs namely, proper max-point-tolerance graphs (proper
MPTG) where intervals associated to the vertices are not contained in each
other properly. We present the first characterization theorem of this graph
class by defining certain linear ordering on the vertex set. In course of this
study we prove proper max-point-tolerance graphs are asteroidal triple free and
perfect. We also find proper max-point-tolerance graphs are equivalent to unit
max-point-tolerance graphs. Further we note that MPTG (proper MPTG) and
max-tolerance graphs (proper max-tolerance graphs) are incomparable. In
conclusion we demonstrate relations between proper MPTG with other variants of
MPTG and max-tolerance graphs.
