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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Fixed Simplex Property for Retractable Complexes | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 303640 7 pages doi 10.1155 2010 303640 Research Article Fixed Simplex Property for Retractable Complexes Adam Idzik1 2 and Anna Zapart3 1 Institute of Mathematics Jan Kochanowski University 15 ổwietokrzyska street 25-406 Kielce Poland 2 Institute of Computer Science Polish Academy of Sciences 21 Ordona street 01-237 Warsaw Poland 3 Faculty of Mathematics and Information Science Warsaw University of Technology Pl. Politechniki 1 00-661 Warsaw Poland Correspondence should be addressed to Adam Idzik adidzik@ipipan.waw.pl Received 16 December 2009 Revised 10 August 2010 Accepted 9 September 2010 Academic Editor L. Gorniewicz Copyright 2010 A. Idzik and A. Zapart. 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. 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 Nesetril theorem any endomorphism of a dismantlable graph fixes some clique. Properties of recursively contractible complexes are examined. 1. Preliminaries We apply some combinatorial methods in the fixed point theory 1 . These methods allow us to extend some known theorems for graphs 2 and to suggest algorithmic procedures finding fixed simplices for simplicial maps defined on some classes of complexes. By N we denote the set of natural numbers. Let V be a finite set and In 0 . n n e N. By P V we denote the family of all nonempty subsets of V and Pn V P n V is the family of all subsets of V of the cardinality n 1 at most n 1 n e N. A subset Hn c P n V is called a .
