Homotopy Theory for Digraphs

Alexander Grigor’yan University of Bielefeld Yong Lin Renmin University of China Yuri Muranov University of Warmia and Mazury Shing-Tung Yau Harvard University

Combinatorics mathscidoc:1804.06001

Pure and Applied Mathematics Quarterly, 10, 619-674, 2014
We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy invariance of homologies of digraphs and the relation between the fundamental group of the digraph and its first homology group. The category of (undirected) graphs can be identified by a natural way with a full subcategory of digraphs. Thus we obtain also consistent homology and homotopy theories for graphs.
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  title={Homotopy Theory for Digraphs},
  author={Alexander Grigor’yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau},
  booktitle={Pure and Applied Mathematics Quarterly},
Alexander Grigor’yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. Homotopy Theory for Digraphs. 2014. Vol. 10. In Pure and Applied Mathematics Quarterly. pp.619-674. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180403140954177187008.
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