On The Abstract Properties Of Markov Graphs For Maps On Trees

Having a dynamical system on the vertex set of a finite tree, one can construct the corresponding Markov graph which is the digraph that encodes covering relation between edges in a tree. Representatives of isomorphism classes of Markov graphs are called M-graphs. In this paper we prove that the class of M-graphs is closed under several prescribed digraph transformations (such as deletion of a vertex in a digraph or taking the disjoint union of digraphs, for example). We also give a complete list of tournaments which are M-graphs as well as of M-graphs with three vertices.