Finite Sholander Trees, Trees, and their Betweenness

We provide a proof of Sholander's claim (Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree. Furthermore, we streamline Burigana's characterization (Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185, 5-36 (2009)) of tree betweenness and provide a relatively short proof.Comment: 8 page

Finite sholander trees, trees, and their betweennesses

We provide a proof of a claim made by Sholander (Trees, lattices, order, and betweenness, {\it Proc. Amer. Math. Soc.} {\bf 3}, 369-381 (1952)) concerning the representability of collections of so-called segments by trees. Furthermore, we strengthen Burigana's axiomatic characterization of so-called betweennesses induced by trees (Tree representations of betweenness relations defined by intersection and inclusion, {\it Mathematics and Social Sciences} {\bf 185}, 5-36 (2009)) and provide a short proof

Guides and Shortcuts in Graphs

The geodesic structure of a graphs appears to be a very rich structure. There are many ways to describe this structure, each of which captures only some aspects. Ternary algebras are for this purpose very useful and have a long tradition. We study two instances: signpost systems, and a special case of which, step systems. Signpost systems were already used to characterize graph classes. Here we use these for the study of the geodesic structure of a spanning subgraph F with respect to its host graph G. Such a signpost system is called a guide to (F,G). Our main results are: the characterization of the step system of a cycle, the characterization of guides for spanning trees and hamiltonian cycles

Transit functions on graphs (and posets)

The notion of transit function is introduced to present a unifying approach for results and ideas on intervals, convexities and betweenness in graphs and posets. Prime examples of such transit functions are the interval function I and the induced path function J of a connected graph. Another transit function is the all-paths function. New transit functions are introduced, such as the cutvertex transit function and the longest path function. The main idea of transit functions is that of ‘transferring’ problems and ideas of one transit function to the other. For instance, a result on the interval function I might suggest similar problems for the induced path function J. Examples are given of how fruitful this transfer can be. A list of Prototype Problems and Questions for this transferring process is given, which suggests many new questions and open problems

Conservative median algebras and semilattices

We characterize conservative median algebras and semilattices by means of forbidden substructures and by providing their representation as chains. Moreover, using a dual equivalence between median algebras and certain topological structures, we obtain descriptions of the median-preserving mappings between products of finitely many chains