Directed Transit Functions

TL;DR

This paper introduces directed transit functions as a new way to model betweenness in directed structures like posets and graphs, generalizing existing concepts and exploring their properties.

Contribution

It defines directed transit functions, provides axiomatic characterizations, and extends classical transit functions to directed settings such as posets and graphs.

Findings

Betweenness in posets can be characterized by simple first-order axioms.

Directed transit functions generalize geometric transit functions to directed structures.

Properties of directed analogues of all-paths, induced paths, and shortest paths are discussed.

Abstract

Transit functions were introduced as models of betweenness on undirected structures. Here we introduce directed transit function as the directed analogue on directed structures such as posets and directed graphs. We first show that betweenness in posets can be expressed by means of a simple set of first order axioms. Similar characterizations can be obtained for graphs with natural partial orders, in particular, forests, trees, and mangroves. Relaxing the acyclicity conditions leads to a generalization of the well-known geometric transit function to the directed structures. Moreover, we discuss some properties of the directed analogues of prominent transit functions, including the all-paths, induced paths, and shortest paths (or interval) transit functions. Finally we point out some open questions and directions for future work.