On subclasses of interval count two and on Fishburn’s conjecture

Abstract

An interval graph is the intersection graph of a family F of intervals on the real line. An interval order is a partial order ( F , ≺ ) , where F is a family of intervals on the real line, such that for all I 1 , I 2 ∈ F , I 1 ≺ I 2 if and only if I 1 lies entirely to the left of I 2 . In both cases, the family F is called a model of the graph (order). The interval count of a given graph (resp. order) is the smallest number of interval lengths needed in any model of this graph (resp. order). The first problem we consider is related to the classes of graphs and orders which can be represented with two interval lengths a and b , with respect to the inclusion hierarchy among such classes for variables a , b . The second problem is an extremal problem which consists of determining the smallest graph or order which has interval count at least k . In particular, we study a conjecture by Fishburn on this extremal problem, verifying its validity when such a conjecture is constrained to the classes of trivially perfect orders and split orders.