Characterization and representation problems for intersection betweennesses

Abstract

For a set system M = ( M v ) v ∈ V indexed by the elements of a finite set V , the intersection betweenness B ( M ) induced by M consists of all triples ( u , v , w ) ∈ V 3 with M u ∩ M w ⊆ M v . Similarly, the strict intersection betweenness B s ( M ) induced by M consists of all triples ( u , v , w ) ∈ B ( M ) such that u , v , and w 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.