Let T be a finite or infinite tree and let V 0 be the set of all vertices of T of valency 1. We propose a sufficient condition for the image of the imbedding ψ: T \V 0 → \( {{\mathbb{R}}^2} \) to be a level set of a pseudoharmonic function.
Similar content being viewed by others
References
M. Morse, Topological Methods in the Theory of Functions of a Complex Variable, Princeton University Press, Princeton (1947).
E. Polulyakh and I. Yurchuk, On the Pseudo-Harmonic Functions Defined on a Disk, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2009).
V. A. Rokhlin and D. B. Fuks, A First Course in Topology. Geometric Chapters [in Russian], Nauka, Moscow (1977).
M. H. A. Newman, Elements of the Topology of Plane Sets of Points, Cambridge Univ. Press, Cambridge (1964).
C. Berge, Théorie des Graphes et Ses Applications, Dunod, Paris (1958).
O. Ore, Theory of Graphs, American Mathematical Society, Providence, RI (1962).
Y. Tôki, “A topological characterization of pseudo-harmonic functions,” Osaka Math. J., 3, No. 1, 101–122 (1951).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 7, pp. 974–995, July, 2013.
Rights and permissions
About this article
Cite this article
Polulyakh, E.O. Trees as Level Sets for Pseudoharmonic Functions in the Plane. Ukr Math J 65, 1087–1110 (2013). https://doi.org/10.1007/s11253-013-0844-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-013-0844-0