Homotopy groups of Hom complexes of graphs

The notion of $\times$-homotopy from \cite{DocHom} is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space \Hom_*(G,H) with the homotopy groups of \Hom_*(G,H^I). Here \Hom_*(G,H) is a space which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the usual \Hom complex), and $H^I$ is the graph of based paths in $H$. As a corollary it is shown that \pi_i \big(\Hom_*(G,H) \big) \cong [G,\Omega^i H]_{\times}, where $\Omega H$ is the graph of based closed paths in $H$ and $[G,K]_{\times}$ is the set of $\times$-homotopy classes of pointed graph maps from $G$ to $K$. This is similar in spirit to the results of \cite{BBLL}, where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed.Comment: 20 pages, 6 figures, final version, to be published in J. Combin. Theory Ser.

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.Comment: 28 pages, 13 figures, final version, to be published in European J. Com

Spider Web Graphs

Recent results from Chih-Scull show that the ×-homotopy relation on finite graphs can be expressed via a sequence of “spider moves” which shift a single vertex at a time. In this paper we study a “spider web graph” which encodes exactly these spider moves between graph homomorphisms. We show how composition of graph homomorphisms relate to the spider web, study the components of spider webs for bipartite and tree graphs, andfinish by giving an explicit description of the spider web for homomorphisms from a bipartite graph to a star graph

Du cadastre ancien au graphe. Les dynamiques spatiales dans les sources fiscales médiévales et modernes

http://archeosciences.revues.org/3758National audienceMedieval and modern tax documents (fieldbooks, "compoix", cadasters...) give a rich spatial information. Whole territories are described plot per plot at different succesive periods. Nevertheless, historians don't know how to relate different states of spatial information on the long term. Moreover, there are no plot plans before the 17th and 18th century. We want to overcome this methodological lock with the following proposition: modelling ancient plots described in tax documents using the topological proprieties of the graph theory. The translation of the spatial data into graphs should allow to set up a commun language between the historical documents, mapped and not mapped. The article talks about the fondamental method choices and the used process to transform the spatial information of ancient plans into graphs which will allow us not only to compare them together, but also to compare them with cadastral registers that have no plan. Finally, it is within a Geographic Information System that will be tested the pertinence of comparing possibilities while studying the plot changes according to the spatiotemporal operators of changement (creation, disappearance, stability, dilatation, contraction, fusion, fission, deformation).Les documents fiscaux médiévaux et modernes (terriers, compoix, cadastres...) offrent une information spatiale riche. Des territoires entiers sont décrits parcelle par parcelle à des époques successives. Les historiens ne parviennent toutefois jamais à corréler ces différents états de l'information spatiale sur la longue durée, d'autant qu'on ne dispose d'aucun plan parcellaire avant les xviie-xviiie siècles. C'est ce verrou méthodologique qu'il s'agit de dépasser en proposant, dans le cadre de l'ANR Modelespace, une modélisation des parcellaires anciens décrits dans les documents fiscaux en mettant en œuvre les propriétés topologiques de la théorie des graphes. La conversion des données spatiales en graphes doit permettre de mettre en place un langage commun entre les documents historiques. L'article traite des choix méthodologiques fondamentaux et de la démarche utilisée pour transformer l'information spatiale des plans anciens en graphes susceptibles d'être appariés non seulement entre eux, mais encore avec des graphes issus de matrices cadastrales sans plan. Au final, c'est au sein d'un Système d'Information Géographique que devra être testée la pertinence des possibilités de comparaisons en étudiant les recompositions parcellaires selon des opérateurs spatio-temporels de changement (création, disparition, stabilité, dilatation, contraction, fusion, fission, déformation)

Incidence Hypergraphs: Box Products & the Laplacian

The box product and its associated box exponential are characterized for the categories of quivers (directed graphs), multigraphs, set system hypergraphs, and incidence hypergraphs. It is shown that only the quiver case of the box exponential can be characterized via homs entirely within their own category. An asymmetry in the incidence hypergraphic box product is rectified via an incidence dual-closed generalization that effectively treats vertices and edges as real and imaginary parts of a complex number, respectively. This new hypergraphic box product is shown to have a natural interpretation as the canonical box product for graphs via the bipartite representation functor, and its associated box exponential is represented as homs entirely in the category of incidence hypergraphs; with incidences determined by incidence-prism mapping. The evaluation of the box exponential at paths is shown to correspond to the entries in half-powers of the oriented hypergraphic signless Laplacian matrix.Comment: 34 pages, 23 figures, 4 table