Geodesic
On disjoint union of $\mathrm{M}$-graphs
Algebra and discrete mathematics, Tome 24 (2017) no. 2, pp. 262-273.

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Given a pair $(X,\sigma)$ consisting of a finite tree $X$ and its vertex self-map $\sigma$ one can construct the corresponding Markov graph $\Gamma(X,\sigma)$ which is a digraph that encodes $\sigma$-covering relation between edges in $X$. $\mathrm{M}$-graphs are Markov graphs up to isomorphism. We obtain several sufficient conditions for the disjoint union of $\mathrm{M}$-graphs to be an $\mathrm{M}$-graph and prove that each weak component of $\mathrm{M}$-graph is an $\mathrm{M}$-graph itself.
Keywords: tree maps, Markov graphs, Sharkovsky's theorem. 
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Sergiy Kozerenko. On disjoint union of $\mathrm{M}$-graphs. Algebra and discrete mathematics, Tome 24 (2017) no. 2, pp. 262-273. http://geodesic.mathdoc.fr/item/ADM_2017_24_2_a6/

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