On characterizing proper max-point-tolerance graphs

Abstract

Max-point-tolerance graphs (MPTG) were introduced by Catanzaro et al. in 2017 as a generalization of interval graphs. This graph class has many practical applications in the study of the human genome as well as in signal processing for networks. The same class of graphs was also studied by Soto and Caro in 2015 with a different name, p -BOX(1) graphs. In our article, we consider a natural subclass of max-point-tolerance graphs, namely, proper max-point-tolerance graphs (proper MPTG), where intervals associated with 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 the course of this study, we prove proper max-point-tolerance graphs are asteroidal triple-free, and perfect. We also find that proper max-point-tolerance graphs are equivalent to unit max-point-tolerance graphs. Further, we show 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.