On the Cut-Vertex and the Interval Transit Functions of Hypergraphs

Abstract

Transit functions R:V×V→2V model abstract betweenness as well as binary clustering. Examples are I(u, v), the interval between u and v, comprising all points on a shortest path from u to v, and C(u, v), the set of all cut vertices separating u and v together with u and v. Here we characterize the cut-vertex transit function of hypergraphs as the monotone transit functions satisfying (x) R(u,v)⊆R(u,x)∪R(x,v) for all u,v,x∈V. We define new hypergraph classes as restrictions and generalizations of linear hypergraphs and describe relevant properties of blocks and Berge cycles. We then show that the cut-vertex transit function coincides with the interval function exactly for linear B∗-hypergraphs, generalizing a similar result for graphs. Moreover, we identify a subclass of block hypergraphs and characterize it using axioms on its interval function and prove a similar characterization for block graphs.