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Abstract: Skeletons are notoriously sensitive to contour noise, and an effective filtering scheme is needed in any practical situation, where skeletons are involved. In this article, we introduce a new discrete framework that allows us to define and compute families of filtered Euclidean skeletons, in 2D as well as in 3D or higher dimensions. We prove several properties of our skeletonization scheme, in particular the preservation of topological characteristics and the stability with respect to parameter changes. [Copyright &y& Elsevier]

Abstract: We describe an algorithm to compute the geodesics in an arbitrary CAT(0) cubical complex. A key tool is a correspondence between cubical complexes of global non-positive curvature and posets with inconsistent pairs. This correspondence also gives an explicit realization of such a complex as the state complex of a reconfigurable system, and a way to embed any interval in the integer lattice cubing of its dimension. [Copyright &y& Elsevier]

Abstract: We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifted cubical complexes, in particular, these eigenvalues are also integers. Finally, we recover Adinʼs enumeration of spanning trees of a complete colorful simplicial complex from the Cellular Matrix-Tree Theorem together with a result of Kook, Reiner and Stanton. [Copyright &y& Elsevier]

Abstract: In the geometric realization of a cubical complex without degeneracies, a □-set, dipaths and dihomotopies may not be combinatorial, i.e., not geometric realizations of combinatorial dipaths and equivalences. When we want to use geometric/topological tools to classify dipaths on the 1-skeleton, combinatorial dipaths, up to dihomotopy, and in particular up to combinatorial dihomotopy, we need that all dipaths are in fact dihomotopic to a combinatorial dipath. And moreover that two combinatorial dipaths which are dihomotopic are then combinatorially dihomotopic. We prove that any dipath from a vertex to a vertex is dihomotopic to a combinatorial dipath, in a non-selfintersecting □-set. And that two combinatorial dipaths which are dihomotopic through a non-combinatorial dihomotopy are in fact combinatorially dihomotopic, in a geometric □-set. Moreover, we prove that in a geometric □-set, the d-homotopy introduced in [M. Grandis, Directed homotopy theory, I, Cah. Topol. Géom. Différ. Catég. 44 (4) (2003) 281–316] coincides with the dihomotopy in [L. Fajstrup, E. Goubault, M. Raussen, Algebraic topology and concurrency, Theoret. Comput. Sci., in press; also technical report, Aalborg University, 1999]. [Copyright &y& Elsevier]