Characterization and representation problems for intersection betweennesses Original Research Article

For a set system image indexed by the elements of a finite set image, the intersection betweenness image induced by image consists of all triples image with image. Similarly, the strict intersection betweenness image induced by image consists of all triples image such that image, image, and image are pairwise distinct. The notion of a strict intersection betweenness was introduced by Burigana [L. Burigana, Tree representations of betweenness relations defined by intersection and inclusion, Math. Soc. Sci. 185 (2009) 5–36]. We provide axiomatic characterizations of intersection betweennesses and strict intersection betweennesses. Our results yield a simple and efficient algorithm that constructs a representing set system for a given (strict) intersection betweenness. We study graphs whose strict shortest path betweenness is a strict intersection betweenness. Finally, we explain how the algorithmic problem related to Burigana’s notion of a partial tree representation can be solved efficiently using well-known algorithms.