Unfoldings and Coverings of Weighted Graphs
Résumé
Coverings of undirected graphs are used in distributed computing, and unfoldings of directed graphs in semantics of programs. We study these two notions from a graph theoretical point of view so as to highlight their similarities, as they are both dened in terms of surjective graph homomorphisms. In particular, universal coverings and complete unfoldings are innite trees that are regular if the initial graphs are nite. Regularity means that a tree has nitely many subtrees up to isomorphism. Two important theorems have been established by Leighton and Norris for coverings of nite graphs. We prove similar results for unfoldings of nite directed graphs. Moreover, we generalize coverings and similarly, unfoldings to graphs and digraphs equipped with nite or innite weights attached to edges of the covered or unfolded graphs. This generalization yields a canonical "factorization" of the universal covering of any nite graph, that (provably) does not exist without using weights. Introducing ω as an innite weight provides us with nite descriptions of regular trees having nodes of countably innite degree. Regular trees (trees having nitely many subtrees up to isomorphism) play an important role in the extension of Formal Language Theory to innite structures described in nitary ways. Our weighted graphs offer effective descriptions of the above mentioned regular trees and yield decidability results. We also generalize to weighted graphs and their coverings a classical factorization theorem of their characteristic polynomials.
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