  A tolled walk $T$ between two non-adjacent vertices $u$ and $v$ in a graph
$G$ is a walk, in which $u$ is adjacent only to the second vertex of $T$ and
$v$ is adjacent only to the second-to-last vertex of $T$. A 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 $S \subseteq V(G)$ is
toll convex, if $T_{G}(u,v)\subseteq S$ for all $u,v\in S$. A toll closure of a
set $S \subseteq V(G)$ is the union of toll intervals between all pairs of
vertices from $S$. The size of a smallest set $S$ whose toll closure is the
whole vertex set is called a toll number of a graph $G$, $tn(G)$. This paper
investigates the toll number of the strong product of graphs. First, a
description of toll intervals between two vertices in the strong product graphs
is given. Using this result we characterize graphs with $tn(G \boxtimes H)=2$
and graphs with $tn(G \boxtimes H)=3$, which are the only two possibilities. As
an addition, for the t-hull number of $G\boxtimes H $ we show that $th(G
\boxtimes H) = 2$ for any non complete graphs $G$ and $H$. As extreme vertices
play an important role in different convexity types, we show that no vertex of
the strong product graph of two non complete graphs is an extreme vertex with
respect to the toll convexity.
