Minimal Triangulations Research Articles

Strongly minimal triangulations of (S 3×S 1)#3 and (S 3 S 1)#3

Strongly minimal triangulations of (S 3×S 1)#3 and (S 3 S 1)#3

Efficiently decomposing, recognizing and triangulating hole-free graphs without diamonds

A graph is hole- and diamond-free (HD-free) if none of its induced subgraphs is isomorphic to a chordless cycle of length at least five or to a diamond. Using the clique separator approach and the simple structure of atoms of HD-free graphs, we show how to recognize HD-free graphs in time O(n2). One of the main tools is Lexicographic Breadth-First Search (LexBFS); we give two new properties of LexBFS which are essential for reaching the time bound and which hold for any graph. Moreover, we find minimal triangulations of HD-free graphs in time O(n2), introducing efficient algorithms for the minimal triangulation of matched co-bipartite graphs and chordal bipartite graphs.

Inflations of ideal triangulations

Starting with an ideal triangulation of M∘, the interior of a compact 3-manifold M with boundary, no component of which is a 2-sphere, we provide a construction, called an inflation of the ideal triangulation, to obtain a strongly related triangulation of M itself. Besides a step-by-step algorithm for such a construction, we provide examples of an inflation of the two-tetrahedra ideal triangulation of the complement of the figure-eight knot in S3, giving a minimal triangulation, having ten tetrahedra, of the figure-eight knot exterior. As another example, we provide an inflation of the one-tetrahedron Gieseking manifold giving a minimal triangulation, having seven tetrahedra, of a non-orientable compact 3-manifold with Klein bottle boundary. Several applications of inflations are discussed.

The Performance of Covariance Selection Methods That Consider Decomposable Models Only

We consider the behavior of Bayesian procedures that perform model selection for decomposable Gaussian graphical models when the true model is in fact non-decomposable. We examine the asymptotic behavior of the posterior when models are misspecified in this way, and find that the posterior will converge to graphical structures that are minimal triangulations of the true structure. The marginal log likelihood ratio comparing different minimal triangulations is stochastically bounded, and appears to remain data dependent regardless of the sample size. The covariance matrices corresponding to the different minimal triangulations are essentially equivalent, so model averaging is of minimal benefit. Using simulated data sets and a particular high performing Bayesian method for fitting decomposable models, feature inclusion stochastic search, we illustrate that these predictions are borne out in practice. Finally, a comparison is made to penalized likelihood methods for graphical models, which make no decomposability restriction. Despite its inability to fit the true model, feature inclusion stochastic search produces models that are competitive or superior to the penalized likelihood methods, especially at higher dimensions.

A New Approach to Crushing 3-Manifold Triangulations

The crushing operation of Jaco and Rubinstein is a powerful technique in algorithmic 3-manifold topology: it enabled the first practical implementations of 3-sphere recognition and prime decomposition of orientable manifolds, and it plays a prominent role in state-of-the-art algorithms for unknot recognition and testing for essential surfaces. Although the crushing operation will always reduce the size of a triangulation, it might alter its topology, and so it requires a careful theoretical analysis for the settings in which it is used. The aim of this short paper is to make the crushing operation more accessible to practitioners, and easier to generalise to new settings. When the crushing operation was first introduced, the analysis was powerful but extremely complex. Here we give a new treatment that reduces the crushing process to a sequential combination of three "atomic" operations on a cell decomposition, all of which are simple to analyse. As an application, we generalise the crushing operation to the setting of non-orientable 3-manifolds, where we obtain a new practical and robust algorithm for non-orientable prime decomposition. We also apply our crushing techniques to the study of non-orientable minimal triangulations.

Organizing the atoms of the clique separator decomposition into an atom tree

We define an atom tree of a graph as a generalization of a clique tree: its nodes are the atoms obtained by clique minimal separator decomposition, and its edges correspond to the clique minimal separators of the graph.Given a graph G, we compute an atom tree by using a clique tree of a minimal triangulation H of G. Computing an atom tree with such a clique tree as input can be done in O(min(nm,m+nf)), where f is the number of fill edges added by the triangulation. When both a minimal triangulation and the clique minimal separators of G are provided, we compute an atom tree of G in O(m+f) time, which is in O(n2) time.We give an O(nm) time algorithm, based on MCS, which combines in a single pass the 3 steps involved in building an atom tree: computing a minimal triangulation, constructing a clique tree, and constructing the corresponding atom tree.Finally, we present a process which uses a traversal of a clique tree of a minimal triangulation to determine the clique minimal separators and build the corresponding atom tree in O(n(n+t)) time, where t is the number of 2-pairs of H (t is at most m¯−f, where m¯ is the number of edges of the complement graph); to complete this, we also give an algorithm which computes a minimal triangulation in O(n(n+m¯)) time, thus providing an approach to compute the decomposition in O(n(n+m¯)) time.

A revisit of the scheme for computing treewidth and minimum fill-in

In this paper, we reformulate the scheme introduced by Bouchitté and Todinca in [1], which computes treewidth and minimum fill-in of a graph using a dynamic programming approach. We will call the scheme BT scheme. Although BT scheme was originally designed for computing treewidth and minimum fill-in, it can be used for computing other graph parameters defined in terms of minimal triangulation. In this paper, we reformulate BT scheme so that it works for computing other graph parameters defined in terms of minimal triangulation, and give examples of other graph parameters.

Algorithms for unipolar and generalized split graphs

A graph G=(V,E) is a unipolar graph if there exists a partition V=V1∪V2 such that V1 is a clique and V2 induces the disjoint union of cliques. The complement-closed class of generalized split graphs contains those graphs G such that either Gor the complement of G is unipolar. Generalized split graphs are a large subclass of perfect graphs. In fact, it has been shown that almost all C5-free (and hence, almost all perfect graphs) are generalized split graphs. In this paper, we present a recognition algorithm for unipolar graphs that utilizes a minimal triangulation of the given graph, and produces a partition when one exists. Our algorithm has running time O(nm+nmF), where mF is the number of edges added in a minimal triangulation of the given graph. Generalized split graphs can be recognized via this algorithm in O(n3) time. We give algorithms on unipolar graphs for finding a maximum independent set and a minimum clique cover in O(n+m) time, and for finding a maximum clique and a minimum proper coloring in O(n2.5/logn) time, when a unipolar partition is given. These algorithms yield algorithms for the four optimization problems on generalized split graphs that have the same worst-case time bounds. We also report that the perfect code problem is NP-complete for chordal unipolar graphs.

Two characterisations of the minimal triangulations of permutation graphs

A minimal triangulation of a graph is a chordal supergraph with an inclusion-minimal edge set. Minimal triangulations are obtained from adding edges only to minimal separators, completing minimal separators into cliques. Permutation graphs are the comparability graphs whose complements are also comparability graphs. Permutation graphs can be characterised as the intersection graphs of specially arranged line segments in the plane, which is called a permutation diagram. The minimal triangulations of permutation graphs are known to be interval graphs, and they can be obtained from permutation diagrams by applying a geometric operation, that corresponds to the completion of separators into cliques. We precisely specify this geometric completion process to obtain minimal triangulations, and we completely characterise those interval graphs that are minimal triangulations of permutation graphs.

Constructing perfect phylogenies and proper triangulations for three-state characters.

In this paper, we study the problem of constructing perfect phylogenies for three-state characters. Our work builds on two recent results. The first result states that for three-state characters, the local condition of examining all subsets of three characters is sufficient to determine the global property of admitting a perfect phylogeny. The second result applies tools from minimal triangulation theory to the partition intersection graph to determine if a perfect phylogeny exists. Despite the wealth of combinatorial tools and algorithms stemming from the chordal graph and minimal triangulation literature, it is unclear how to use such approaches to efficiently construct a perfect phylogeny for three-state characters when the data admits one. We utilize structural properties of both the partition intersection graph and the original data in order to achieve a competitive time bound.

$$\mathbb Z _2$$ -Thurston norm and complexity of 3-manifolds

A new lower bound on the complexity of a 3-manifold is given using the \({\mathbb Z }_2\)-Thurston norm. This bound is shown to be sharp, and the minimal triangulations realising it are characterised using normal surfaces consisting entirely of quadrilateral discs.

Coverings and minimal triangulations of 3–manifolds

This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space $L(4k, 2k-1)$ and the generalised quaternionic space $S^3/Q_{4k}$ have complexity $k,$ where $k\ge 2.$ Moreover, it is shown that their minimal triangulations are unique.

Fast minimal triangulation algorithm using minimum degree criterion

We propose an algorithm for minimal triangulation which, using simple and efficient strategy, subdivides the input graph in different, almost non-overlapping, subgraphs. Using the technique of matrix multiplication for saturating the minimal separators, we show that the partition of the graph can be computed in time O ( n α ) where n α is the time required by the binary matrix multiplication. After saturating the minimal separators, the same procedure is recursively applied on each subgraphs. We also present a variant of the algorithm in which the minimum degree criterion is used. In this way, we obtain an algorithm that uses minimum degree criterion and at the same time produces a minimal triangulation, thus shedding new light on the effectiveness of the minimum degree heuristics.

Convex polyhedra with triangular faces and cone triangulation

Considering the problem of the minimal triangulation for a given polyhedra (dividing polyhedra into tetrahedra) it is known that the cone triangulation provides the number of tetrahedra which is the smallest, or the closest to it. It is also shown that when we want to know whether the cone triangulation is the minimal one, it is necessary to find the order of all vertices, as well as the order of ?separating circles?. Here, we will give algorithms for testing the necessary condition for the cone triangulation if it is the minimal one. The algorithm for forming the cone triangulation will also be given.

Treewidth and minimum fill-in on permutation graphs in linear time

Permutation graphs form a well-studied subclass of cocomparability graphs. Permutation graphs are the cocomparability graphs whose complements are also cocomparability graphs. A triangulation of a graph G is a graph H that is obtained by adding edges to G to make it chordal. If no triangulation of G is a proper subgraph of H then H is called a minimal triangulation. The main theoretical result of the paper is a characterisation of the minimal triangulations of a permutation graph, that also leads to a succinct and linear-time computable representation of the set of minimal triangulations. We apply this representation to devise linear-time algorithms for various minimal triangulation problems on permutation graphs, in particular, we give linear-time algorithms for computing treewidth and minimum fill-in on permutation graphs.

Combinatorial Properties of the K3 Surface: Simplicial Blowups and Slicings

The 4-dimensional abstract Kummer variety K 4 with 16 nodes leads to the K3 surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of K 4, we resolve its 16 isolated singularities—step by step—by simplicial blowups. As a result we obtain a 17-vertex triangulation of the standard PL K3 surface. A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from real projective 3-space onto the 2-sphere with the minimum number of vertices. Moreover, we study simplicial Morse functions and the changes of their levels between the critical points. In this way we obtain slicings through the K3 surface of various topological types.

The Multi-State Perfect Phylogeny Problem with Missing and Removable Data: Solutions via Integer-Programming and Chordal Graph Theory

The Multi-State Perfect Phylogeny Problem is an extension of the Binary Perfect Phylogeny Problem, allowing characters to take on more than two states. In this article, we consider three problems that extend the utility of the multi-state perfect phylogeny model: (1) the Missing Data (MD) Problem, where some entries in the input are missing and the question is whether (bounded) values for the missing data can be imputed so that the resulting data has a multi-state perfect phylogeny; (2) the Character-Removal (CR) Problem, where we want to minimize the number of characters to remove from the data so that the resulting data has a multi-state perfect phylogeny; and (3) the Missing-Data Character-Removal (MDCR) Problem, where the input has missing data and we want to impute values for the missing data to minimize the solution to the resulting Character-Removal Problem. We discuss Integer Linear Programming (ILP) solutions to these problems for the special case of three, four, and five permitted states per character, and we report on extensive empirical testing of these solutions. Then we develop a general theory to solve the MD problem for an arbitrary number of permitted states, using chordal graph theory and results on minimal triangulation of non-chordal graphs. This establishes new necessary and sufficient conditions for the existence of a perfect phylogeny with (or without) missing data. We implement the general theory using integer linear programming, although other optimization methods are possible. We extensively explore the empirical behavior of the general solution, showing that the methods are very practical for data of size and complexity that is characteristic of many current applications in phylogenetics. Some of the empirical results for the MD problem with an arbitrary number of permitted states are very surprising, suggesting the existence of additional combinatorial structure in multi-state perfect phylogenies. Finally, we note some relationships between our chordal-graph approach to the multi-state perfect phylogeny, without missing data, and prior methods.

A note on minimal d-separation trees for structural learning

Structural learning of a Bayesian network is often decomposed into problems related to its subgraphs, although many approaches without decomposition were proposed. In 2006, Xie, Geng and Zhao proposed using a d-separation tree to improve the power of conditional independence tests and the efficiency of structural learning. In our research note, we study a minimal d-separation tree under a partial ordering, by which the maximal efficiency can be obtained. Our results demonstrate that a minimal d-separation tree of a directed acyclic graph (DAG) can be constructed by searching for the clique tree of a minimal triangulation of the moral graph for the DAG.

Simple algorithms for minimal triangulation of a graph and backward selection of a decomposable Markov network

In this paper we propose a simple algorithm called CliqueMinTriang for computing a minimal triangulation of a graph. If F is the set of edges that is added to G to make it a complete graph K n then the asymptotic complexity of CliqueMinTriang is O ( | F | ( δ 2 + | F | ) ) where δ is the degree of the subgraph of K n induced by F . Therefore our algorithm performs well when G is a dense graph. We also show how to exploit the existing minimal triangulation techniques in conjunction with CliqueMinTriang to efficiently find a minimal triangulation of nondense graphs. Finally we show how the algorithm can be adapted to perform a backward stepwise selection of decomposable Markov networks; the resulting procedure has the same time complexity as that of existing similar algorithms.

Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers

We show that no minimal vertex triangulation of a closed, connected, orientable 2-manifold of genus 6 admits a polyhedral embedding in ℝ3. We also provide examples of minimal vertex triangulations of closed, connected, orientable 2-manifolds of genus 5 that do not admit any polyhedral embeddings. Correcting a previous error in the literature, we construct the first infinite family of such nonrealizable triangulations of surfaces. These results were achieved by transforming the problem of finding suitable oriented matroids into a satisfiability problem. This method can be applied to other geometric realizability problems, e.g., for face lattices of polytopes.