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arXiv · 2310.20237

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

Manoj Changat·Jeny Jacob·Lekshmi Kamal K. Sheela·Iztok Peterin
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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.

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Manoj Changat, Jeny Jacob, Lekshmi Kamal K. Sheela, Iztok Peterin. 2023-10-31. The Toll Walk Transit Function of a Graph: Axiomatic Characterizations and First-Order Non-definability. https://arxiv.org/abs/2310.20237

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