\(s\)-homotopy for finite graphs. (English) Zbl 1267.05074
Ossona de Mendez, Patrice (ed.) et al., The international conference on topological and geometric graph theory. Papers from the conference (TGGT 2008) held at the École Normale Supérieure, Paris, France, May 19–23, 2008. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 31, 123-127 (2008).
Summary: We introduce the notion of “\(s\)-dismantlability” which will give an analogue of formal deformations defining the simple-homotopy type in the category of finite simplicial complexes in the category of finite graphs. 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 J. A. Barmak and E. G. Minian [Adv. Math. 218, No. 1, 87–104 (2008; Zbl 1146.57034)] within the framework of posets (theorem 4.2).
For the entire collection see [Zbl 1239.05009].
For the entire collection see [Zbl 1239.05009].
MSC:
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
Citations:
Zbl 1146.57034References:
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| [3] | Chen, B.; Yau, S.-T.; Yeh, Y.-N., Graph homotopy and Graham homotopy, Discrete Math., 241, 1-3, 153-170 (2001) · Zbl 0990.05120 |
| [4] | Cohen, M. M., (A course in simple-homotopy theory. A course in simple-homotopy theory, Graduate Texts in Mathematics, Vol. 10 (1973), Springer-Verlag: Springer-Verlag New York-Berlin) · Zbl 0261.57009 |
| [5] | Ginsburg, J., Dismantlability revisited for ordered sets and graphs and the fixed clique property, Canad. Math. Bull., 37, 4, 473-481 (1994) · Zbl 0824.05057 |
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