Opuscula Math. 41, no. 1 (2021), 55-70
https://doi.org/10.7494/OpMath.2021.41.1.55

Opuscula Mathematica

# More on linear and metric tree maps

Sergiy Kozerenko

Abstract. We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.

Keywords: tree, Markov graph, metric map, non-expanding map, linear map, graph homomorphism.

Mathematics Subject Classification: 05C05, 05C12, 05C20, 54E40.

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• Communicated by Andrzej Żak.
• Revised: 2020-12-23.
• Accepted: 2020-12-28.
• Published online: 2021-02-08. 