Intersection properties of line graphs

AbstractEach graph is an intersection graph (intersection multigraph) of a family of sets. Such a family is called a representation if all sets are different and a pseudorepresentation if some are the same. By intersection number we mean the cardinality of the smallest set on which we can construct a representation. So each graph has four intersection numbers. A graph is uniquely representable if it has only one representation on the minimal set. We take into account only triangle free graphs which are uniquely intersectable in any sense and line graphs. We show that line graphs with pendant K4 - e and clepsydras are line graphs which are not uniquely multiple pseudointersectable (u.m.p.i.) in the class of line graphs