Fast Recognition of Partial Star Products and Quasi Cartesian Products

This paper is concerned with the fast computation of a relation $\R$ on the edge set of connected graphs that plays a decisive role in the recognition of approximate Cartesian products, the weak reconstruction of Cartesian products, and the recognition of Cartesian graph bundles with a triangle free basis. A special case of $\R$ is the relation $\delta^\ast$, whose convex closure yields the product relation $\sigma$ that induces the prime factor decomposition of connected graphs with respect to the Cartesian product. For the construction of $\R$ so-called Partial Star Products are of particular interest. Several special data structures are used that allow to compute Partial Star Products in constant time. These computations are tuned to the recognition of approximate graph products, but also lead to a linear time algorithm for the computation of $\delta^\ast$ for graphs with maximum bounded degree. Furthermore, we define \emph{quasi Cartesian products} as graphs with non-trivial $\delta^\ast$. We provide several examples, and show that quasi Cartesian products can be recognized in linear time for graphs with bounded maximum degree. Finally, we note that quasi products can be recognized in sublinear time with a parallelized algorithm

Fast recognition of partial star products and quasi cartesian products

This paper is concerned with the fast computation of a relation d on the edge set of connected graphs that plays a decisive role in the recognition of approximate Cartesian products, the weak reconstruction of Cartesian products, and the recognition of Cartesian graph bundles with a triangle free basis. A special case of d is the relation , whose convex closure yields the product relation that induces the prime factor decomposition of connected graphs with respect to the Cartesian product. For the construction of d so-called Partial Star Products are of particular interest. Several special data structures are used that allow to compute Partial Star Products in constant time. These computations are tuned to the recognition of approximate graph products, but also lead to a linear time algorithm for the computation of for graphs with maximum bounded degree. Furthermore, we define quasi Cartesian products as graphs with non-trivial . We provide several examples, and show that quasi Cartesian products can be recognized in linear time for graphs with bounded maximum degree. Finally, we note that quasi products can be recognized in sublinear time with a parallelized algorithm.Web of Science115212411

Geometry Optimization of Crystals by the Quasi-Independent Curvilinear Coordinate Approximation

The quasi-independent curvilinear coordinate approximation (QUICCA) method [K. N\'emeth and M. Challacombe, J. Chem. Phys. {\bf 121}, 2877, (2004)] is extended to the optimization of crystal structures. We demonstrate that QUICCA is valid under periodic boundary conditions, enabling simultaneous relaxation of the lattice and atomic coordinates, as illustrated by tight optimization of polyethylene, hexagonal boron-nitride, a (10,0) carbon-nanotube, hexagonal ice, quartz and sulfur at the $\Gamma$-point RPBE/STO-3G level of theory.Comment: Submitted to Journal of Chemical Physics on 7/7/0

Isometric Embeddings in Trees and Their Use in Distance Problems

International audienceWe present powerful techniques for computing the diameter, all the eccentricities, and other related distance problems on some geometric graph classes, by exploiting their "tree-likeness" properties. We illustrate the usefulness of our approach as follows: (1) We propose a subquadratic-time algorithm for computing all eccentricities on partial cubes of bounded lattice dimension and isometric dimension O(n^{0.5−ε}). This is one of the first positive results achieved for the diameter problem on a subclass of partial cubes beyond median graphs. (2) Then, we obtain almost linear-time algorithms for computing all eccentricities in some classes of face-regular plane graphs, including benzenoid systems, with applications to chemistry. Previously, only a linear-time algorithm for computing the diameter and the center was known (and an O(n^{5/3})-time algorithm for computing all the eccentricities). (3) We also present an almost linear-time algorithm for computing the eccentricities in a polygon graph with an additive one-sided error of at most 2. (4) Finally, on any cube-free median graph, we can compute its absolute center in almost linear time. Independently from this work, Bergé and Habib have recently presented a linear-time algorithm for computing all eccentricities in this graph class (LAGOS'21), which also implies a linear-time algorithm for the absolute center problem. Our strategy here consists in exploiting the existence of some embeddings of these graphs in either a system or a product of trees, or in a single tree but where each vertex of the graph is embedded in a subset of nodes. While this may look like a natural idea, the way it can be done efficiently, which is our main technical contribution in the paper, is surprisingly intricate

Combinatorial models for topology-based geometric modeling

Many combinatorial (topological) models have been proposed in geometric modeling, computational geometry, image processing or analysis, for representing subdivided geometric objects, i.e. partitionned into cells of different dimensions: vertices, edges, faces, volumes, etc. We can distinguish among models according to the type of cells (regular or not regular ones), the type of assembly ("manifold" or "non manifold"), the type of representation (incidence graphs or ordered models), etc