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Mat. Zametki, 2005, Volume 78, Issue 2, Pages 223–233 (Mi mz2576)  

This article is cited in 2 scientific papers (total in 2 papers)

A metric of constant curvature on polycycles

M. Dezaa, M. I. Shtogrinb

a Ècole Normale Supérieure, Département de mathématiques et applications
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We prove the following main theorem of the theory of $(r,q)$-polycycles. Suppose a nonseparable plane graph satisfies the following two conditions:
  • 1) each internal face is an r-gon, where $r\ge3$;
  • 2) the degree of each internal vertex is $q$, where $q\ge3$, and the degree of each boundary vertex is at most $q$ and at least 2.
Then it also possesses the following third property:
  • 3) the vertices, the edges, and the internal faces form a cell complex.
Simple examples show that conditions 1) and 2) are independent even provided condition 3) is satisfied. These are the defining conditions for an $(r,q)$-polycycle.

DOI: https://doi.org/10.4213/mzm2576

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English version:
Mathematical Notes, 2005, 78:2, 204–212

Bibliographic databases:

UDC: 514.17+519.17
Received: 08.05.2003
Revised: 01.04.2004

Citation: M. Deza, M. I. Shtogrin, “A metric of constant curvature on polycycles”, Mat. Zametki, 78:2 (2005), 223–233; Math. Notes, 78:2 (2005), 204–212

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. Deza, M. I. Shtogrin, “Archimedean polycycles”, Russian Math. Surveys, 59:3 (2004), 564–566  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. M. Deza, S. V. Shpektorov, M. I. Shtogrin, “Non-extendible finite polycycles”, Izv. Math., 70:1 (2006), 1–18  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  • Математические заметки Mathematical Notes
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