Path-induced closure operators on graphs for defining digital Jordan surfaces. (English) Zbl 1513.54128
Summary: Given a simple graph with the vertex set \(X\), we discuss a closure operator on \(X\) induced by a set of paths with identical lengths in the graph. We introduce a certain set of paths of the same length in the 2-adjacency graph on the digital line \(\mathbb{Z}\) and consider the closure operators on \(\mathbb{Z}^m (m\) a positive integer) that are induced by a special product of \(m\) copies of the introduced set of paths. We focus on the case \(m = 3\) and show that the closure operator considered provides the digital space \(\mathbb{Z}^3\) with a connectedness that may be used for defining digital surfaces satisfying a Jordan surface theorem.
MSC:
| 54H30 | Applications of general topology to computer science (e.g., digital topology, image processing) |
| 54A05 | Topological spaces and generalizations (closure spaces, etc.) |
| 54D05 | Connected and locally connected spaces (general aspects) |
| 52C99 | Discrete geometry |
| 68R10 | Graph theory (including graph drawing) in computer science |
Keywords:
simple graph; path; closure operator; connectedness; digital space; digital surface; Khalimsky topology; Jordan surface theoremReferences:
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