In this article, we study model structures on the category of finite graphs with \(\times \)-homotopy equivalences as the weak equivalences. We show that there does not exist an analogue of Strøm-Hurewicz model structure on this category of graphs. More interestingly, we show that this category of graphs with \(\times \)-homotopy equivalences does not have a model structure whenever the class of cofibrations is a subclass of graph inclusions.

中文翻译：

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图类别上的（缺乏）模型结构

在本文中，我们以\（\ times \） -同等性为弱等价性来研究有限图类上的模型结构。我们证明在此类图上不存在Strøm-Hurewicz模型结构的类似物。更有趣的是，我们证明，当共纤化类别是图包含物的子类时，具有\（\ times \） -同等性的这类图不具有模型结构。