arXiv subjects

Éric Colin de Verdière

Publications and source records attributed to Éric Colin de Verdière.

20 records · Page 2Linked to original sources

arXiv2013

Testing Graph Isotopy on Surfaces

We investigate the following problem: Given two embeddings G_1 and G_2 of the same abstract graph G on an orientable surface S, decide whether G_1 and G_2 are isotopic; in other words, whether there exists a continuous family of embeddings between G_1 and G_2. We provide efficient algorithms to solve this problem in two models. In the first model, the input consists of the arrangement of G_1 (resp., G_2) with a fixed graph cellularly embedded on S; our algorithm is linear in the input complexity, and thus, optimal. In the second model, G_1 and G_2 are piecewise-linear embeddings in the plane minus a finite set of points; our algorithm runs in O(n^{3/2}\log n) time, where n is the complexity of the input. The graph isotopy problem is a natural variation of the homotopy problem for closed curves on surfaces and on the punctured plane, for which algorithms have been given by various authors; we use some of these algorithms as a subroutine. As a by-product, we reprove the following mathematical characterization, first observed by Ladegaillerie (1984): Two graph embeddings are isotopic if and only if they are homotopic and congruent by an oriented homeomorphism.

cs.CG↗

arXiv2011

Helly numbers of acyclic families

The Helly number of a family of sets with empty intersection is the size of its largest inclusion-wise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Gamma. Assume that for every sub-family G of F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(d_Gamma+1), where d_Gamma is the smallest integer j such that every open set of Gamma has trivial Q-homology in dimension j and higher. (In particular d_{R^d} = d). This bound is best possible. We prove, in fact, a stronger theorem where small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known.

math.CO↗