Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Homomorphism complexes, reconfiguration, and homotopy for directed
  graphs   The neighborhood complex of a graph was introduced by Lov\'asz to provide
topological lower bounds on chromatic number, and more general homomorphism
complexes of graphs were further studied by Babson and Kozlov. Such `Hom
complexes' are also related to reconfiguration problems and a notion of
discrete homotopy for graphs. Here we initiate the detailed study of Hom
complexes for directed graphs, which have applications in the study of graded
posets and resolutions of monomial ideals. Our construction can be seen as a
special case of the poset structure on the set of multihomomorphisms in more
general categories, as introduced by Kozlov, Matsushita, and others. We relate
the topological properties of Hom complexes to certain digraph operations
including products, adjunctions, and foldings. We introduce a notion of a
neighborhood complex for a directed graph and prove that its homotopy type is
recovered as a certain Hom complex. We establish a number of results regarding
the topology of these complexes, including the dependence on directed bipartite
subgraphs, a directed version of the Mycielksi construction, as well as
vanishing theorems for higher homology. Inspired by notions of reconfigurations
of directed graph colorings we study the connectivity of Hom complexes into
tournaments $T_n$. We obtain a complete answer for the case of transitive
$T_n$, where we also describe a connection to mixed subdivisions of dilated
simplices. Finally we use paths in the internal hom objects of directed graphs
to define various notions of homotopy, and discuss connections to the topology
of Hom complexes.

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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin