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We study continuous countably (strictly) monotone maps defined on a tame graph, i.e. a special Peano continuum for which the set containing branch points and end points has 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 map 
$g$ of constant slope. In particular, we show that in the case of a Markov map 
$f$ that corresponds to a recurrent transition matrix, the condition is satisfied for a constant slope 
$e^{h_{\text{top}}(f)}$, where 
$h_{\text{top}}(f)$ is the topological entropy of 
$f$. Moreover, we show that in our class the topological entropy 
$h_{\text{top}}(f)$ is achievable through horseshoes of the map 
$f$.
