On uniquely intersectable graphs Original Research Article

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.