Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Toll number of the Cartesian and the lexicographic product of graphs   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.

11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess
16
The categories of this paper are: stat
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
14
The categories of this paper are: math-ph
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th