Minimal Triangulations Research Articles

Monomial ideals whose powers have a linear resolution

In this paper we consider graded ideals in a polynomial ring over a field and ask when such an ideal has the property that all of its powers have a linear resolution. Recall that a graded module M is said to have a linear resolution if all entries in the matrices representing the differentials in a graded minimal free resolution of M are linear forms. If an ideal I has a linear resolution, then necessarily all generators of I have the same degree, say t . In that case, one also says that I has a t-linear resolution. It is known [7] that polymatroidal ideals have linear resolutions and that powers of polymatroidal ideals are again polymatroidal (see [2] and [8]). In particular they have again linear resolutions. In general however, powers of ideals with linear resolution need not to have linear resolutions. The first example of such an ideal was given by Terai. He showed that over a base field of characteristic = 2 the Stanley Reisner ideal I = (abd, abf, ace, adc, aef, bde, bcf, bce, cdf, def ) of the minimal triangulation of the projective plane has a linear resolution, while I 2 has no linear resolution. The example depends on the characteristic of the base field. If the base field has characteristic 2, then I itself has no linear resolution. Another example, namely I = (def, cef, cdf, cde, bef, bcd, acf, ade) is given by Sturmfels [13]. Again I has a linear resolution, while I 2 has no linear resolution. The example of Sturmfels is interesting because of two reasons: 1. it does not depend on the characteristic of the base field, and 2. it is a linear quotient ideal. Recall that an equigenerated ideal I is said to have linear quotients if there exists an order f1, . . . , fm of the generators of I such that for all i = 1, . . . , m the colon ideals (f1, . . . , fi−1) : fi are generated by linear forms. It is quite easy to see that such an ideal has a linear resolution (independent on the characteristic of the base field). However the example of Sturmfels also shows that powers of a linear quotient ideal need not to be again

Maximum Cardinality Search for Computing Minimal Triangulations of Graphs

We present a new algorithm, called MCS-M, for computing minimal triangulations of graphs. Lex-BFS, a seminal algorithm for recognizing chordal graphs, was the genesis for two other classical algorithms: LEX M and MCS. LEX M extends the fundamental concept used in Lex-BFS, resulting in an algorithm that not only recognizes chordality, but also computes a minimal triangulation of an arbitrary graph. MCS simplifies the fundamental concept used in Lex-BFS, resulting in a simpler algorithm for recognizing chordal graphs. The new algorithm MCS-M combines the extension of LEX M with the simplification of MCS, achieving all the results of LEX M in the same time complexity.

0-Efficient Triangulations of 3-Manifolds

0-efficient triangulations of 3-manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3-manifold M can be modified to a 0-efficient triangulation or M can be shown to be one of the manifolds S3,ℝP3 or L(3, 1). Similarly, any triangulation of a compact, orientable, irreducible, ∂-irreducible 3-manifold can be modified to a 0-efficient triangulation. The notion of a 0-efficient ideal triangulation is defined. It is shown if M is a compact, orientable, irreducible, ∂-irreducible 3-manifold having no essential annuli and distinct from the 3-cell, then \ring M admits an ideal triangulation; furthermore, it is shown that any ideal triangulation of such a 3-manifold can be modified to a 0-efficient ideal triangulation. A 0-efficient triangulation of a closed manifold has only one vertex or the manifold is S3 and the triangulation has precisely two vertices. 0-efficient triangulations of 3-manifolds with boundary, and distinct from the 3-cell, have all their vertices in the boundary and then just one vertex in each boundary component. As tools, we introduce the concepts of barrier surface and shrinking, as well as the notion of crushing a triangulation along a normal surface. A number of applications are given, including an algorithm to construct an irreducible decomposition of a closed, orientable 3-manifold, an algorithm to construct a maximal collection of pairwise disjoint, normal 2-spheres in a closed 3-manifold, an alternate algorithm for the 3-sphere recognition problem, results on edges of low valence in minimal triangulations of 3-manifolds, and a construction of irreducible knots in closed 3-manifolds.

Combining polynomial running time and fast convergence for the disk-covering method

This paper treats polynomial-time algorithms for reconstruction of phylogenetic trees. The disc-covering method (DCM) presented by Huson et al. (J. Comput. Biol. 6 (3/4) (1999) 369) is a method that boosts the performance of phylogenetic tree construction algorithms. Actually, they gave two variations of DCM-Buneman. The first variation was guaranteed to recover the true tree with high probability from polynomial-length sequences (i.e. polynomial in the number of given taxa), but it was not proven to run in polynomial time. The second variation was guaranteed to run in polynomial time. However, it is a heuristic in the sense that it was not proven to recover the true tree with high probability from polynomial-length sequences. We present an improved DCM. The difference between our improved DCM and the heuristic variation of the original DCM is marginal. The main contribution of this paper is the analysis of the algorithm. Our analysis shows that the improved DCM combines the desirable properties of the two variations of the original DCM. That is, it runs in polynomial time and it recovers the true tree with high probability from polynomial-length sequences. Moreover, this is true when the improved DCM is applied to the Neighbor-Joining, the Buneman, as well as the Agarwala algorithm. A key observation for the result of Huson et al. was that threshold graphs of additive distance functions are chordal. We prove a chordal graph theorem concerning minimal triangulations of threshold graphs constructed from distance functions which are close to being additive. This theorem is the key observation behind our improved DCM and it may be interesting in its own right.

Listing all potential maximal cliques of a graph

A potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph. It is known that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We show here that the potential maximal cliques of a graph can be generated in polynomial time in the number of minimal separators of the graph. Thus, the treewidth and the minimum fill-in are polynomially tractable for all classes of graphs with a polynomial number of minimal separators.

Asteroidal triples of moplexes

An asteroidal triple is an independent set of vertices such that each pair is joined by a path that avoids the neighborhood of the third, and a moplex is an extension to an arbitrary graph of a simplicial vertex in a triangulated graph. The main result of this paper is that the investigation of the set of moplexes of a graph is sufficient to conclude as to its having an asteroidal triple. Specifically, we show that a graph has an asteroidal triple of vertices if and only if it has an asteroidal triple of moplexes. We also examine the behavior of an asteroidal triple of moplexes in the course of a minimal triangulation process, and give some related properties.

On the Structure of Graphs with Bounded Asteroidal Number

A set A⊆V of the vertices of a graph G=(V,E) is an asteroidal set if for each vertex a∈A, the set A\{a} is contained in one component of G−N[a]. The maximum cardinality of an asteroidal set of G, denoted by an (G), is said to be the asteroidal number of G. We investigate structural properties of graphs of bounded asteroidal number. For every k≥1, an (G)≤k if and only if an (H)≤k for every minimal triangulation H of G. A dominating target is a set D of vertices such that D∪S is a dominating set of G for every set S such that G[D∪S] is connected. We show that every graph G has a dominating target with at most an (G) vertices. Finally, a connected graph G has a spanning tree T such that d T (x,y)−d G (x,y)≤3·|D|−1 for every pair x,y of vertices and every dominating target D of G.

Moplex elimination orderings

Abstract Classically, triangulated graphs are characterized and recognized by way of perfect elimination orderings (peo, which correspond to an elimination scheme on simplicial vertices). Algorithm LexBFS computes such a peo efficiently, but is also useful for the enumeration of the minimal separators and maximal cliques, which means that the ordering it produces is special. We present an ordering which generalizes a LexBFS-type ordering, by eliminating at each step a set of simplicial vertices (called a moplex). This type of ordering actually characterizes triangulated graphs and easily yields the minimal separators and maximal cliques. We establish a strong relationship between moplex elimination orderings and clique trees. We also extend our results to the problem of embedding an arbitrary graph into a triangulated graph by showing that this process of computing a minimal triangulation is essentially equivalent to forcing a graph into having a simplicial moplex elimination ordering

Treewidth and Minimum Fill-in: Grouping the Minimal Separators

We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We prove that for all classes of graphs for which polynomial algorithms computing the treewidth and the minimum fill-in exist, we can list their potential maximal cliques in polynomial time. Our approach unifies these algorithms. Finally we show how to compute in polynomial time the potential maximal cliques of weakly triangulated graphs for which the treewidth and the minimum fill-in problems were open.

Approximation for Minimum Triangulations of Simplicial Convex 3-Polytopes

A minimum triangulation of a convex 3-polytope is a triangulation that contains the minimum number of tetrahedra over all its possible triangulations. Since finding minimum triangulations of convex 3-polytopes was recently shown to be NP-hard, it becomes significant to find algorithms that give good approximation. In this paper we give a new triangulation algorithm with an improved approximation ratio 2 - Ω(1/\sqrt n ) , where n is the number of vertices of the polytope. We further show that this is the best possible for algorithms that only consider the combinatorial structure of the polytopes.

Finding Small Triangulations of Polytope Boundaries Is Hard

We prove that it is NP-hard to decide whether a polyhedral 3-ball can be triangulated with k simplices. The construction also implies that it is difficult to find the minimal triangulation of such a 3-ball. A lifting argument is used to transfer the result also to triangulations of boundaries of 4-polytopes. The proof is constructive and translates a variant of the 3-SAT problem into an instance of a concrete polyhedral 3-ball for which it is difficult to find a minimal triangulation.

A Minimal Triangulation of the Hopf Map and its Application

We give a minimal triangulation η: S 12 3 →S 4 2 of the Hopf map h: S 3→S 2 and use it to obtain a new construction of the 9-vertex complex projective plane.

Minimal Simplicial Dissections and Triangulations of Convex 3-Polytopes

This paper addresses three questions related to minimal triangulations of a three-dimensional convex polytope P .

On the structure of graphs with bounded asteroidal number

A set A⊆V of the vertices of a graph G=(V,E) is an asteroidal set if for each vertex a∈A, the set A\{a} is contained in one component of G−N[a]. The maximum cardinality of an asteroidal set of G, denoted by an (G), is said to be the asteroidal number of G. We investigate structural properties of graphs of bounded asteroidal number. For every k≥1, an (G)≤k if and only if an (H)≤k for every minimal triangulation H of G. A dominating target is a set D of vertices such that D∪S is a dominating set of G for every set S such that G[D∪S] is connected. We show that every graph G has a dominating target with at most an (G) vertices. Finally, a connected graph G has a spanning tree T such that d T (x,y)−d G (x,y)≤3·|D|−1 for every pair x,y of vertices and every dominating target D of G.

New results on MWT subgraphs

Let P be a polygon in the plane. We disprove the conjecture that the so-called LMT-skeleton coincides with the intersection of all locally minimal triangulations, LMT( P), even for convex polygons P. We introduce an improved LMT-skeleton algorithm which, for any simple polygon P, exactly computes LMT( P), and thus a larger subgraph of the minimum-weight triangulation MWT( P). The algorithm achieves the same in the general point set case provided the connectedness of the improved LMT-skeleton, which is given in almost all practical instances. We further observe that the β-skeleton of P is a subset of MWT( P) for all values β > 4 3 provided P is convex or near-convex. This gives evidence for the tightness of this bound in the general point set case.

Asymptotics for the length of a minimal triangulation on a random sample

Given $F \subset [0, 1]^2$ and finite, let $\sigma(F)$ denote the length of the minimal Steiner triangulation of points in F. By showing that minimal Steiner triangulations fit into the theory of subadditive and superadditive Euclidean functionals, we prove under a mild regularity condition that $$\lim_{n \to \infty} \sigma(X_1,\dots, X_n)/n^{1/2} = \beta \int_{[0, 1]^2}f(x)^{1/2} dx \c.c.,$$ where $X_1,\dots, X_n$ are i.i.d. random variables with values in $[0, 1]^2$, $\beta$ is a positive constant, f is the density of the absolutely continuous part of the law of $X_1$ , and c.c. denotes complete convergence. This extends the work of Steele. The result extends naturally to dimension three and describes the asymptotics for the probabilistic Plateau functional, thus making progress on a question of Beardwood, Halton and Hammersley. Rates of convergence are also found.

Characterizations and algorithmic applications of chordal graph embeddings

We introduce the separator graph for a given graph G and show a 1-1 correspondence between its maximal cliques and the minimal triangulations (i.e., ⊆-minimal chordal embeddings) of G. This approach can be used for characterizations of graph classes by properties of their minimal separators. In particular, we show that a graph is AT-free if and only if every minimal triangulation is an interval graph, that a graph is claw-free AT-free if and only if every minimal triangulation is a proper interval graph, and that a graph is a cograph if and only if every minimal triangulation is a trivially perfect graph. These results have algorithmic consequences for several graph parameters that are related to triangulation problems. In this context, we also show how the vertex ranking problem can be formulated as a triangulation problem into trivially perfect graphs. As consequences for the claw-free AT-free graphs we obtain that the bandwidth equals the treewidth and pathwidth, and that the proper interval completion number equals the chordal completion number and interval completion number. This directly implies that computing the bandwidth or interval completion number is NP-hard even for co-bipartite graphs and, on the other side, that there are efficient algorithms for these problems on many other claw-free subclasses of co-comparability graphs.

A Large Subgraph of the Minimum Weight Triangulation

We present an O(n 4 )-time and O(n 2 )-space algorithm that computes a subgraph of the minimum weight triangulation (MWT) of a general point set. The algorithm works by finding a collection of edges guaranteed to be in any locally minimal triangulation. We call this subgraph the LMT-skeleton. We also give a variant called the modified LMT-skeleton that is both a more complete subgraph of the MWT and is faster to compute requiring only O(n 2 ) time and O(n) space in the expected case for uniform distributions. Several experimental implementations of both approaches have shown that for moderate-sized point sets (up to 350 points^1) the skeletons are connected, enabling an efficient completion of the exact MWT. We are thus able to compute the MWT of substantially larger random point sets than have previously been computed. ^1Though in this paper we summarize some empirical findings for input sets of up to 350 points, a variant of the algorithm has been implemented and tested on up to 40,000 points producing connected subgraphs [2].

Asteroidal Triple-Free Graphs

An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triple-free (AT-free) if it contains no asteroidal triples. The motivation for this investigation was provided, in part, by the fact that the AT-free graphs provide a common generalization of interval, permutation, trapezoid, and cocomparability graphs. The main contribution of this work is to investigate and reveal fundamental structural properties of AT-free graphs. Specifically, we show that every connected AT-free graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. We then provide characterizations of AT-free graphs in terms of dominating pairs and minimal triangulations. Subsequently, we state and prove a decomposition theorem for AT-free graphs. An assortment of other properties of AT-free graphs is also provided. These properties generalize known structural properties of interval, permutation, trapezoid, and cocomparability graphs.

Suppressing curvature fluctuations in dynamical triangulations

We study numerically the dynamical triangulation formulation of two-dimensional quantum gravity using a restricted class of triangulation, so-called minimal triangulations, in which only vertices of coordination number 5, 6 and 7 are allowed [1]. A real-space RG analysis shows that for pure gravity (central charge c = 0) this restriction does not affect the critical behavior of the model. Furthermore, we show that the critical behavior of an Ising model coupled to minimal dynamical triangulations ( c = 1 2 ) is still governed by the KPZ-exponents.