Almtract--We present a comprehensive survey of the theory of hypercube graphs. Basic properties related to distance, coloring, domination and genus are reviewed. The properties of the n-cube defined by its subgraphs are considered next, including thickness, coarseness, Hamiltonian cycles and induced paths and cycles. Finally, various embedding and packing problems are discussed, including the d...

Cohomology and cohomology ring of three-dimensional (3D) objects are topological invariants that characterize holes and their relations. Cohomology ring has been traditionally computed on simplicial complexes. Nevertheless, cubical complexes deal directly with the voxels in 3D images, no additional triangulation is necessary, facilitating efficient algorithms for the computation of topological ...

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, pr...

Abstract This article proposes a way of doing type theory informally, assuming cubical style reasoning. It can thus be viewed as first step toward alternative to the program informalization carried out in homotopy book for dependent augmented with axioms univalence and higher inductive types. We adopt cartesian proposed by Angiuli, Brunerie, Coquand, Favonia, Harper, Licata implicit foundation,...

We study finite foldable cubical complexes of nonpositive curvature (in the sense of A.D. Alexandrov). We show that such a complex X admits a graph of spaces decomposition. It is also shown that when dimX = 3, X contains a closed rank one geodesic in the 1-skeleton unless the universal cover of X is isometric to the product of two CAT(0) cubical complexes. AMS Classification 20F65, 20F67; 53C20

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 cubi...

The cubical sets model of Homotopy Type Theory was introduced by Bezem, Coquand and Huber using a particular category of presheaves. We show that this category is equivalent to a category of sets whose elements have a finite support property with respect to an action of a monoid of name substitutions; and that this is isomorphic to a category of nominal sets equipped with source and target maps...