Open Access

Fixed Simplex Property for Retractable Complexes

Fixed Point Theory and Applications20102010:303640

https://doi.org/10.1155/2010/303640

Received: 16 December 2009

Accepted: 9 September 2010

Published: 14 September 2010

Abstract

Retractable complexes are defined in this paper. It is proved that they have the fixed simplex property for simplicial maps. This implies the theorem of Wallace and the theorem of Rival and Nowakowski for finite trees: every simplicial map transforming vertices of a tree into itself has a fixed vertex or a fixed edge. This also implies the Hell and Nešetřil theorem: any endomorphism of a dismantlable graph fixes some clique. Properties of recursively contractible complexes are examined.

Publisher note

To access the full article, please see PDF

Authors’ Affiliations

(1)
Institute of Mathematics, Jan Kochanowski University
(2)
Institute of Computer Science, Polish Academy of Sciences
(3)
Faculty of Mathematics and Information Science, Warsaw University of Technology

Copyright

© Adam Idzik and Anna Zapart. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.