Directed Transit Functions

TL;DR

The paper generalizes transit functions to directed structures, providing a clean axiomatic framework that recovers posets via directed betweenness and extends to DAGs, rooted forests/trees, and general graphs through reachability and path-based constructions. It establishes a tight correspondence between poset functions and the induced partial orders, and shows that directed transit functions on graphs yield natural notions such as reachability, all-paths, induced-path, and interval functions, each with varying degrees of geometric structure and axiom independence. Key contributions include a complete poset-function characterization, a transitive-reduction perspective for DAGs, and a detailed survey of directed counterparts to classic undirected transit-function notions, including new results on the geometric and interval cases. The work lays a foundation for directed betweenness modeling in networks and points to several open questions, such as first-order characterizations for all-directed transit functions and the graphic realizability of directed interval functions.

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.

Figures (5)

• Figure 1: The reachability function of this digraph satisfies $R_{\to}(x_1,x_2)=\{u,v,w\}$ for any two distinct vertices $x_1,x_2\in\{u,v,w\}$. Thus $G_{R_{\to}}$ is edge-less and hence $G\not\cong G_{R_{\to}}$.
• Figure 2: Example of a directed graph that does not satisfy many of the axioms that are essential to prove the existence of paths for DAGs, or their posets. For example we do not have $R(u,w)\neq\emptyset$ and $R(w,v)\neq\emptyset$ implies that $w\in R(u,v)$, hence (tr0) is not satisfied. Similarly, we have that $R(u,w)\cup R(w,v)\not\subseteq R(u,v)$ and (tr1) does not hold. Since $R(u,w)\cap R(w,v)=\{w,x\}$ also (q) is not satisfied.
• Figure 3: The all-path transit function $A_G$ of the graphs violates one of the betweenness axioms. (a) (b1$_1$) is violated since $x\in R(u,v)=\{u,x,v\}$ but $v\in R(u,x)=\{u,v,x\}$. (b) (b1$_2$) is violated by $x\in R(u,v)=\{u,x,v\}$ and $u\in R(x,v)=\{u,x,v\}$. (c) (b2$_1$) does not hold since $x\in R(u,v)=\{u,x,v\}$ and $w\in R(u,x)=\{u,v,w,x\}$ but $w\notin R(u,v)$ and hence $R(u,x)\not\subseteq R(u,v)$. (d) Analogously, (b2$_2$) is violated by $x\in R(u,v)$ and $w\in R(x,v)$ but $w\notin R(u,v)$.
• Figure 4: A graph $G$ for which the transit function $J_G$ does not satisfy (b5): we have $J_G(u,w)=\{u,...,w\}$, $J_G(u,w)=\{w,...,v\}$, but $J_G(u,v)=\{u,v\}$.
• Figure 5: Examples of graphs for which $J_G$ is not geometric/weakly geometric.

Theorems & Definitions (90)

• Definition 1
• Lemma 1
• proof
• proof
• Definition 2
• Lemma 3
• proof
• Definition 3
• Theorem 4
• proof