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This means that all 1-blocks must be sequences of adjacent 0- and 1-cells, and all junctions between 1-blocks must be 0-blocks. Another approach to finite topological spaces originates from the field of graph theory – the combinatorial map [15, 5]: Definition 4: A combinatorial map is a triple (D, σ, α) where D is a set of darts (also known as half-edges), σ is a permutation of the darts, and α is an involution of the darts. In this context, a permutation is a mapping that associates to each dart a unique predecessor and a unique successor.

Thus, faces in a combinatorial map may not have holes. g. [15]. This allows the realization of non-orientable manifolds and the definition of generalized maps (G-maps) that can be generalized to arbitrary dimensions [12]. However, the half-edge definition suffices in the present context. XPMaps and Topological Segmentation 25 The final approach we are going to deal with is Khalimsky’s grid [6, 3]. It is defined as the product topology of two 1-dimensional topological spaces. In particular, one defines a connected ordered topological space (COTS) as a finite ordered set of points which alternate being open and closed.

18] S. Fourey and R. Malgouyres. Intersection Number of Paths Lying on a Digital Surface and a New Jordan Theorem. In G. Bertrand, M. Couprie, and L. Perroton, editors, Discrete Geometry for Computer Imagery, DGCI’99, volume Vol. 1568 of Lecture Notes in Computer Science, pages 104–117, Marne-la-Vall´ee, France, 1999. Springer, Berlin Heidelberg, New York. [19] R. Glantz and W. G. Kropatsch. Plane Embedding of Dually Contracted Graphs. In G. Borgefors, I. Nystr¨ om, and G. Sanniti di Baja, editors, Proceedings DGCI’00, Discrete Geometry for Computer Imagery, volume Vol.

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