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Galois connections between sets of paths and closure operators in simple graphs. (English) Zbl 1407.05070

Summary: For every positive integer \(n\),we introduce and discuss an isotone Galois connection between the sets of paths of lengths \(n\) in a simple graph and the closure operators on the (vertex set of the) graph. We consider certain sets of paths in a particular graph on the digital line \(\mathbb Z\) and study the closure operators associated, in the Galois connection discussed, with these sets of paths. We also focus on the closure operators on the digital plane \(\mathbb Z^{2}\) associated with a special product of the sets of paths considered and show that these closure operators may be used as background structures on the plane for the study of digital images.

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C38 Paths and cycles
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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