### On disjoint union of \(\mathrm{M}\)-graphs

#### Abstract

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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