The toll walk transit function of a graph: Axiomatic characterizations and first-order non-definability

A walk $W=w_1{w}_2\dots w_k$, $k\ge 2$, is called a toll walk if $w_1\ne w_k$ and $w_2\left(w_{k-1}\right)$ are the only neighbors of $w_1\left(w_k\right)$ on $W$ in a graph $G$. A toll walk interval $T(u,v)$, $u,v\in V\left(G\right)$, contains all the vertices that belong to a toll walk between $u$ and $v$. The toll walk intervals yield a toll walk transit function $T:V\left(G\right)\times V\left(G\right)\to 2^{V\left(G\right)}$. 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 cannot be described in the language of first-order logic for an arbitrary graph.