Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hom complexes and homotopy theory in the category of graphs   We investigate a notion of $\times$-homotopy of graph maps that is based on
the internal hom associated to the categorical product in the category of
graphs. It is shown that graph $\times$-homotopy is characterized by the
topological properties of the $\Hom$ complex, a functorial way to assign a
poset (and hence topological space) to a pair of graphs; $\Hom$ complexes were
introduced by Lov\'{a}sz and further studied by Babson and Kozlov to give
topological bounds on chromatic number. Along the way, we also establish some
structural properties of $\Hom$ complexes involving products and exponentials
of graphs, as well as a symmetry result which can be used to reprove a theorem
of Kozlov involving foldings of graphs. Graph $\times$-homotopy naturally leads
to a notion of homotopy equivalence which we show has several equivalent
characterizations. We apply the notions of $\times$-homotopy equivalence to the
class of dismantlable graphs to get a list of conditions that again
characterize these. We end with a discussion of graph homotopies arising from
other internal homs, including the construction of `$A$-theory' associated to
the cartesian product in the category of reflexive graphs.

15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph