  We study continuous countably (strictly) monotone maps defined on a tame
graph, i.e., a special Peano continuum for which the set containing
branchpoints and endpoints has a countable closure. In our investigation we
confine ourselves to the countable Markov case. We show a necessary and
sufficient condition under which a locally eventually onto, countably Markov
map $f$ of a tame graph $G$ is conjugate to a constant slope map $g$ of a
countably affine tame graph. In particular, we show that in the case of a
Markov map $f$ that corresponds to recurrent transition matrix, the condition
is satisfied for constant slope $e^{h_{\operatorname{top}}(f)}$, where
$h_{\operatorname{top}}(f)$ is the topological entropy of $f$. Moreover, we
show that in our class the topological entropy $h_{\operatorname{top}}(f)$ is
achievable through horseshoes of the map $f$.
