Strong Euler well-composedness

In this paper, we define a new flavour of well-composedness, called strong Euler well-composedness. In the general setting of regular cell complexes, a regular cell complex of dimension n is strongly Euler well-composed if the Euler characteristic of the link of each boundary cell is 1, which is the Euler characteristic of an $$(n-1)$$ ( n - 1 ) -dimensional ball. Working in the particular setting of cubical complexes canonically associated with $$n$$ n D pictures, we formally prove in this paper that strong Euler well-composedness implies digital well-composedness in any dimension $$n\ge 2$$ n ≥ 2 and that the converse is not true when $$n\ge 4$$ n ≥ 4 .

Min-Max Theory for Cell Complexes

In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. We introduce a discretized version of this theory associated to a discrete Morse function on a (regular) cell complex. As applications we prove a discrete version of the mountain pass lemma and give an alternate proof of a discrete Lusternik–Schnirelmann theorem.

Coxeter-like complexes

International audience Motivated by the Coxeter complex associated to a Coxeter system (W,S), we introduce a simplicial regular cell complex Δ (G,S) with a G-action associated to any pair (G,S) where G is a group and S is a finite set of generators for G which is minimal with respect to inclusion. We examine the topology of Δ (G,S), and in particular the representations of G on its homology groups. We look closely at the case of the symmetric group S_n minimally generated by (not necessarily adjacent) transpositions, and their type-selected subcomplexes. These include not only the Coxeter complexes of type A, but also the well-studied chessboard complexes.

THE VISIBILITY COMPLEX

We introduce the visibility complex (a 2-dimensional regular cell complex) of a collection of n pairwise disjoint convex obstacles in the plane. It can be considered as a subdivision of the set of free rays (i.e., rays whose origins lie in free space, the complement of the obstacles). Its cells correspond to collections of rays with the same backward and forward views. The combinatorial complexity of the visibility complex is proportional to the number k of free bitangents of the collection of obstacles. We give an O(n log n+k) time and O(k) working space algorithm for its construction. Furthermore we show how the visibility complex can be used to compute the visibility polygon from a point in O(m log n) time, where m is the size of the visibility polygon. Our method is based on the notions of pseudotriangle and pseudo-triangulation, introduced in this paper.