The main contribution of this paper is a new "extrinsic" digital fundamental group that can be readily generalized to define higher homotopy groups for arbitrary digital spaces. We show that the digital fundamental group of a digital object is naturally isomorphic to the fundamental group of its continuous analogue. In addition, we state a digital version of the Seifert-Van Kampen theorem.
Abstract. In this paper a notion of lighting function is introduced as an axiomatized formalization of the "face membership ruleg' suggested by Kovalevsky. These functions are defined in the context of the framework for digital topology previously developed by the authors. This enlarged framework provides the (a, fl)-connectedness (~, fl E {6, 18, 26}) defined on 2[ 3 within the graph-based approach to digital topology. Furthermore, the Kong-Roscoe (c~,fl)-surfaces, with (a,~) 5~ (6,6), (18,6), are also found as particular cases of a more general notion of digital surface.
Abstract. We show that determining whether or not a simplicial 2− complex collapses to a point is deterministic polynomial time decidable. We do this by solving the problem of constructively deciding whether a simplicial 2−complex collapses to a 1−complex. We show that this proof cannot be extended to the 3D case, by proving that deciding whether a simplicial 3−complex collapses to a 1−complex is an NP −complete problem.
The goal of this paper is to determine the components of the complement of digital manifolds in the standard cubical decomposition of Euclidean spaces for arbitrary dimensions. Our main result generalizes the Morgenthaler-Rosenfeld's one for (26, 6)-surfaces in 7/3 [9]. The proof of this generalization is based on a new approach to digital topology sketched in [5] and developed in [2].
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