Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find

Drawing together techniques from combinatorics and computer science, we improve the census algorithm for enumerating closed minimal P^2-irreducible 3-manifold triangulations. In particular, new constraints are proven for face pairing graphs, and pruning techniques are improved using a modification of the union-find algorithm. Using these results we catalogue all 136 closed non-orientable P^2-irreducible 3-manifolds that can be formed from at most ten tetrahedra.Comment: 37 pages, 34 figure

Testing Linear Inequalities of Subgraph Statistics

Property testers are fast randomized algorithms whose task is to distinguish between inputs satisfying some predetermined property ? and those that are far from satisfying it. Since these algorithms operate by inspecting a small randomly selected portion of the input, the most natural property one would like to be able to test is whether the input does not contain certain forbidden small substructures. In the setting of graphs, such a result was obtained by Alon et al., who proved that for any finite family of graphs ?, the property of being induced ?-free (i.e. not containing an induced copy of any F ? ?) is testable. It is natural to ask if one can go one step further and prove that more elaborate properties involving induced subgraphs are also testable. One such generalization of the result of Alon et al. was formulated by Goldreich and Shinkar who conjectured that for any finite family of graphs ?, and any linear inequality involving the densities of the graphs F ? ? in the input graph, the property of satisfying this inequality can be tested in a certain restricted model of graph property testing. Our main result in this paper disproves this conjecture in the following strong form: some properties of this type are not testable even in the classical (i.e. unrestricted) model of graph property testing. The proof deviates significantly from prior non-testability results in this area. The main idea is to use a linear inequality relating induced subgraph densities in order to encode the property of being a pseudo-random graph

Simplification of UML/OCL schemas for efficient reasoning

Ensuring the correctness of a conceptual schema is an essential task in order to avoid the propagation of errors during software development. The kind of reasoning required to perform such task is known to be exponential for UML class diagrams alone and even harder when considering OCL constraints. Motivated by this issue, we propose an innovative method aimed at removing constraints and other UML elements of the schema to obtain a simplified one that preserve the same reasoning outcomes. In this way, we can reason about the correctness of the initial artifact by reasoning on a simplified version of it. Thus, the efficiency of the reasoning process is significantly improved. In addition, since our method is independent from the reasoning engine used, any reasoning method may benefit from it.Peer ReviewedPostprint (author's final draft

Asymptotics in directed exponential random graph models with an increasing bi-degree sequence

Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we provide for the first time a rigorous analysis of directed exponential random graph models using the in-degrees and out-degrees as sufficient statistics with binary as well as continuous weighted edges. We establish the uniform consistency and the asymptotic normality for the maximum likelihood estimate, when the number of parameters grows and only one realized observation of the graph is available. One key technique in the proofs is to approximate the inverse of the Fisher information matrix using a simple matrix with high accuracy. Numerical studies confirm our theoretical findings.Comment: Published at http://dx.doi.org/10.1214/15-AOS1343 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org