Toll number of the Cartesian and the lexicographic product of graphs

TL;DR

This paper investigates toll convexity in graph theory, specifically analyzing the toll number and toll hull number for Cartesian and lexicographic graph products, providing characterizations and formulas for these parameters.

Contribution

It characterizes toll convex sets in Cartesian products and derives formulas and conditions for the toll number in lexicographic products of graphs.

Findings

Toll number of Cartesian product graphs is always 2.

Toll number of lexicographic product graphs is at most three times that of the first graph.

Graphs with toll number 2 in lexicographic products have a universal vertex and toll number 2 in the second graph.

Abstract

Toll convexity is a variation of the so-called interval convexity. A tolled walk $T$ between $u$ and $v$ in $G$ is a walk of the form $T: u,w_1,\ldots,w_k,v,$ where $k\ge 1$, in which $w_1$ is the only neighbor of $u$ in $T$ and $w_k$ is the only neighbor of $v$ in $T$. As in geodesic or monophonic convexity, toll interval between $u,v\in V(G)$ is a set $T_G(u,v)=\{x\in V(G)\,:\,x \textrm{ lies on a tolled walk between } u \textrm{ and } v\}$. A set of vertices $S$ is toll convex, if $T_{G}(u,v)\subseteq S$ for all $u,v\in S$. First part of the paper reinvestigates the characterization of convex sets in the Cartesian product of graphs. Toll number and toll hull number of the Cartesian product of two arbitrary graphs is proven to be 2. The second part deals with the lexicographic product of graphs. It is shown that if $H$ is not isomorphic to a complete graph, $tn(G \circ H) \leq 3\cdot tn(G)$. We give some necessary and sufficient conditions for $tn(G \circ H) = 3\cdot tn(G)$. Moreover, if $G$ has at least two extreme vertices, a complete characterization is given. Also graphs with $tn(G \circ H)=2$ are characterized - this is the case iff $G$ has an universal vertex and $tn(H)=2$. Finally, the formula for $tn(G \circ H)$ is given - it is described in terms of the so-called toll-dominating triples.