Weakly toll convexity in graph products

The exploration of weakly toll convexity is the focus of this investigation. A weakly toll walk is any walk $W: u, w_1, \ldots , w_{k-1}, v$ between $u$ and $v$ such that $u$ is adjacent only to the vertex $w_1$, which can appear more than once in the walk, and $v$ is adjacent only to the vertex $w_{k-1}$, which can appear more than once in the walk. Through an examination of general graphs and an analysis of weakly toll intervals in both lexicographic and (generalized) corona product graphs, precise values of the weakly toll number for these product graphs are obtained. Notably, in both instances, the weakly toll number is constrained to either 2 or 3. Additionally, the determination of the weakly toll number for the Cartesian and the strong product graphs is established through previously established findings in toll convexity theory. Lastly for all graph products examined within our scope, the weakly toll hull number is consistently determined to be 2

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

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.Comment: 31 pages, 4 figures, 25 reference