s-homotopy for finite graphs |
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| Institution: | 1. Department of Mathematics, University of Illinois, Urbana, IL 61801, USA;2. Bolyai Institute, University of Szeged, Szeged, Hungary;3. University of Warwick, UK;4. Department of Mathematics, Iowa State University, Ames, IA, USA;5. Department of Mathematical and Statistical Sciences, University of Colorado Denver, USA;1. Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland;2. Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland;1. Department of Mathematics, University of South Carolina, Columbia, SC 29212, USA;2. Visiting Professor, Department of Pure and Applied Mathematics, University of Johannesburg, P.O. Box 524, Auckland Park, Johannesburg 2006, South Africa;3. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA |
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| Abstract: | We introduce the notion of “s-dismantlability” which will give in the category of finite graphs an analogue of formal deformations defining the simple-homotopy type in the category of finite simplicial complexes. More precisely, s-dismantlability allows us to define an equivalence relation whose equivalence classes are called “s-homotopy types” and we get a correspondence between s-homotopy types in the category of graphs and simple-homotopy types in the category of simplicial complexes by the way of classical functors between these two categories (theorem 3.6). Next, we relate these results to similar results obtained by Barmak and Minian (2006) within the framework of posets (theorem 4.2). |
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