On uniquely intersectable graphs

In 1977, Alter and Wang (Uniquely intersectable graphs, Discrete Math. 18 (1977) 217-226) introduced the concept of unique intersectability of a graph. They showed that triangle-free is a sufficient condition for a graph to be uniquely intersectable. In 1990, Tsuchiya (On intersection graphs with respect to antichains (II), Utilities Math. 37 (1996) 29-44) studied the concept of unique intersectability with respect to antichains and showed that triangle-free is also a sufficient condition for a graph to be uniquely intersectable with respect to antichains. In this paper we generalize the above results by proving that if a graph is diamond-free and twins-free, then it is uniquely intersectable and if a graph is diamond-free and nonpendant brothers-free, then it is uniquely intersectable with respect to antichains. Also we characterize diamond-free graphs that are uniquely intersectable and the line graphs of triangle-free graphs that are uniquely intersectable. We also consider the concept of unique intersectability with respect to multifamilies and obtain a characterization of such graphs. (C) 1999 Elsevier Science B.V. All rights reserved.