The Toll Walk Transit Function of a Graph: Axiomatic Characterizations and First-Order Non-definability

TL;DR

This paper introduces the toll walk transit function in graphs, characterizes it axiomatically across various graph classes, and proves its non-definability in first-order logic for general graphs.

Contribution

It provides axiomatic characterizations of the toll walk transit function for specific graph classes and establishes its first-order non-definability in arbitrary graphs.

Findings

Axioms characterize toll walk transit in chordal graphs, trees, and others.

Toll walk intervals form a transit function with specific properties.

Non-definability in first-order logic for general graphs.

Abstract

A walk $W=w_1w_2\dots w_k$, $k\geq 2$, is called a toll walk if $w_1\neq w_k$ and $w_2$ and $w_{k-1}$ are the only neighbors of $w_1$ and $w_k$, respectively, on $W$ in a graph $G$. A toll walk interval $T(u,v)$, $u,v\in V(G)$, contains all the vertices that belong to a toll walk between $u$ and $v$. The toll walk intervals yield a toll walk transit function $T:V(G)\times V(G)\rightarrow 2^{V(G)}$. We represent several axioms that characterize the toll walk transit function among chordal graphs, trees, asteroidal triple-free graphs, Ptolemaic graphs, and distance hereditary graphs. We also show that the toll walk transit function can not be described in the language of first-order logic for an arbitrary graph.