We present different ways of endowing a particular category of graphs with Quillen model structures. We show, among other things, that the core of a graph can be seen as its homotopy type in an appropriate Quillen model structure, and that an infinity of Quillen model structures exist for our particular category of graphs.
Homology, Homotopy and Applications, Vol. 14 (2012), No. 2, pp.265-284.
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