Maximal Graphs Research Articles

Geometric Biplane Graphs I: Maximal Graphs

We study biplane graphs drawn on a finite planar point set $S$ in general position. This is the family of geometric graphs whose vertex set is $S$ and can be decomposed into two plane graphs. We show that two maximal biplane graphs---in the sense that no edge can be added while staying biplane---may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over $n$-element point sets.

Maximal surface equation on a Riemannian 2-manifold with finite total curvature

The differential equation of maximal surfaces on a complete Riemannian 2-manifold with finite total curvature is studied. Uniqueness theorems that widely extend the classical Calabi–Bernstein’s theorem in non-parametric version, as well as previous results on complete maximal graphs into Lorentzian warped products, are given. All entire solutions of maximal equation in certain natural Lorentzian warped product, as well as non-existence results, are provided.

Well-posedness of general boundary-value problems for scalar conservation laws

In this paper we investigate well-posedness for the problem u t + div ⁡ φ ( u ) = f u_t+ \operatorname {div} \varphi (u)=f on ( 0 , T ) × Ω (0,T)\!\times \!\Omega , Ω ⊂ R N \Omega \subset \mathbb {R}^N , with initial condition u ( 0 , ⋅ ) = u 0 u(0,\cdot )=u_0 on Ω \Omega and with general dissipative boundary conditions φ ( u ) ⋅ ν ∈ β ( t , x ) ( u ) \varphi (u)\cdot \nu \in \beta _{(t,x)}(u) on ( 0 , T ) × ∂ Ω (0,T)\!\times \!\partial \Omega . Here for a.e. ( t , x ) ∈ ( 0 , T ) × ∂ Ω (t,x)\in (0,T)\!\times \!\partial \Omega , β ( t , x ) ( ⋅ ) \beta _{(t,x)}(\cdot ) is a maximal monotone graph on R \mathbb {R} . This includes, as particular cases, Dirichlet, Neumann, Robin, obstacle boundary conditions and their piecewise combinations. As for the well-studied case of the Dirichlet condition, one has to interpret the formal boundary condition given by β \beta by replacing it with the adequate effective boundary condition. Such effective condition can be obtained through a study of the boundary layer appearing in approximation processes such as the vanishing viscosity approximation. We claim that the formal boundary condition given by β \beta should be interpreted as the effective boundary condition given by another monotone graph β ~ \tilde \beta , which is defined from β \beta by the projection procedure we describe. We give several equivalent definitions of entropy solutions associated with β ~ \tilde \beta (and thus also with β \beta ). For the notion of solution defined in this way, we prove existence, uniqueness and L 1 L^1 contraction, monotone and continuous dependence on the graph β \beta . Convergence of approximation procedures and stability of the notion of entropy solution are illustrated by several results.

Maximal harmonic group actions on finite graphs

This paper studies groups of maximal size acting harmonically on a finite graph. Our main result states that these maximal graph groups are exactly the finite quotients of the modular group Γ=〈x,y∣x2=y3=1〉 of size at least 6. This characterization may be viewed as a discrete analogue of the description of Hurwitz groups as finite quotients of the (2,3,7)-triangle group in the context of holomorphic group actions on Riemann surfaces. In fact, as an immediate consequence of our result, every Hurwitz group is a maximal graph group, and the final section of the paper establishes a direct connection between maximal graphs and Hurwitz surfaces via the theory of combinatorial maps.

The Spectral Radius for a Class of Double-Star-Like Tree Systems with Maximal Degree 4

We mainly study the properties of the 4-double-star-like tree, which is the generalization of star-like trees. Firstly we use graft transformation method to obtain the maximal and minimum extremal graphs of 4-double-star-like trees. Secondly, by the relations between the degree and second degree of vertices in maximal extremal graphs of 4-double-star-like trees we get the upper bounds of spectral radius of 4-double-star-like trees.

Extremal unicyclic graphs with respect to additively weighted Harary index

In this paper we define cycle-star graphCSk;n k to be a graph onn vertices consisting of the cycle of lengthk andn k leafs appended to the same vertex of the cycle. Also, we define cycle-path graphCPk;n k to be a graph onn vertices consisting of the cycle of lengthk and of path on n k vertices whose one end is linked to a vertex on a cycle. We establish that cycle- star graphCS3;n 3 is the only maximal graph with respect to additively weighted Harary index among all unicyclic graphs on n vertices, while cycle-path graph CP3;n 3 is the only minimal unicyclic graph (here n must be at least 5): The values of additively weighted Harary index for extremal unicyclic graphs are established, so these values are the upper and the lower bound for the value of additively weighted Harary index on the class of unicyclic graphs onn vertices. 2010 Mathematics Subject Classification: 05C35

1-Planarity of Graphs with a Rotation System

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. 1-planarity is known NP-hard, even for graphs of bounded bandwidth, pathwidth, or treewidth, and for near-planar graphs in which an edge is added to a planar graph. On the other hand, there is a linear time 1-planarity testing algorithm for maximal 1-planar graphs with a given rotation system. In this work, we show that 1-planarity remains NP-hard even for 3-connected graphs with (or without) a rotation system. Moreover, the crossing number problem remains NP-hard for 3-connected 1-planar graphs with (or without) a rotation system.

Clique-width of full bubble model graphs

Bubble models are 2-dimensional representations of proper interval graphs. We consider proper interval graphs that have bubble models of specific properties. We characterise the maximal such proper interval graphs of bounded clique-width and of bounded linear clique-width and the minimal such proper interval graphs whose clique-width and linear clique-width exceed the bounds. As a consequence, we can efficiently compute the clique-width and linear clique-width of the considered graphs.

A generalization of the Darcy–Forchheimer equation involving an implicit, pressure-dependent relation between the drag force and the velocity

We study mathematical properties of steady flows described by the system of equations generalizing the classical porous media models of Darcy and Forchheimer. The considered generalizations are outlined by implicit relations between the drag force and the velocity, that are in addition parametrized by the pressure. We analyze such drag force–velocity relations which are described through a maximal monotone graph varying continuously with the pressure. Large-data existence of a solution to this system is established, whereupon we show that under certain assumptions on data, the pressure satisfies a maximum or minimum principle, even if the drag coefficient depends on the pressure exponentially.

On-line maximum matching in complete multi-partite graphs with an application to optical networks

Finding a maximum matching in a graph is a classical problem. The on-line versions of the problem in which the vertices and/or edges of the graph are given one at a time and an algorithm has to calculate a matching incrementally have been studied for more than two decades. Many variants of the problem are considered in the literature. The pioneering work (Karp, 1990) considers a bipartite graph where the vertices of one part are revealed one at a time together with their incident edges. In this work we consider maximal d-colorable graphs which are exactly the complete d-partite graphs. The vertices arrive one at a time together with their incident edges, or equivalently with their corresponding colors in some given d-coloring of the graph. We present an optimal 2/3-competitive deterministic algorithm for this on-line problem.This problem is closely related to that of minimizing the cost of line terminals in star topology optical network. We consider lightpaths arriving in an on-line fashion on a given star network. Our result implies a tight 10/9-competitive algorithm for finding a wavelength assignment minimizing the cost of line terminals in such a network.

Stochastic porous media equations in [formula omitted

Existence and uniqueness of solutions to the stochastic porous media equation dX−Δψ(X)dt=XdW in Rd are studied. Here, W is a Wiener process, ψ is a maximal monotone graph in R×R such that ψ(r)≤C|r|m, ∀r∈R. In this general case, the dimension is restricted to d≥3, the main reason being the absence of a convenient multiplier result in the space H={φ∈S′(Rd);|ξ|(Fφ)(ξ)∈L2(Rd)}, for d≤2. When ψ is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space H−1(Rd). If ψ(r)r≥ρ|r|m+1 and m=d−2d+2, we prove the finite time extinction with strictly positive probability.

The number of the maximal triangle-free graphs

Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the previously known lower bound. Our proof uses among others the Ruzsa-Szemer\'{e}di triangle removal lemma, and recent results on characterizing of the structure of independent sets in hypergraphs.

Mining maximal frequent patterns in a single graph using inexact matching

Despite being a popular problem during the last years, frequent graph mining has some important research lines still open for further exploration; among them, the possibility of using inexact matching in order to discover patterns otherwise missed, and the aim at reducing the set of mined patterns to facilitate their analysis and use. We present an algorithm that mines maximal patterns in a single graph using inexact matching. By allowing structural differences, in vertices as well as in edges, between a pattern and its occurrences, our algorithm is able to identify different, often larger, patterns than those found by other state-of-the-art algorithms. On the other hand, by focusing on maximal graphs the output set is significantly downsized without losing patterns, as they can be recovered from the maximal ones. Experiments show that the “extra” patterns found by our algorithm can help in supervised tasks like classification, thus suggesting that useful information about the input data can be captured through the maximal patterns mined with inexact matching.

Markov properties for mixed graphs

In this paper, we unify the Markov theory of a variety of different types of graphs used in graphical Markov models by introducing the class of loopless mixed graphs, and show that all independence models induced by $m$-separation on such graphs are compositional graphoids. We focus in particular on the subclass of ribbonless graphs which as special cases include undirected graphs, bidirected graphs, and directed acyclic graphs, as well as ancestral graphs and summary graphs. We define maximality of such graphs as well as a pairwise and a global Markov property. We prove that the global and pairwise Markov properties of a maximal ribbonless graph are equivalent for any independence model that is a compositional graphoid.

Uniform infinite planar quadrangulations with a boundary

AbstractWe introduce and study the uniform infinite planar quadrangulation (UIPQ) with a boundary via an extension of the construction of Curien et al. (Curien et al., Lat Am J Probab Math Stat 49 (2013) 340–373). We then relate this object to its simple boundary analog using a pruning procedure. This enables us to study the aperture of these maps, that is, the maximal graph distance between two points on the boundary, which in turn sheds new light on the geometry of the UIPQ. In particular we prove that the self‐avoiding walk on the UIPQ is at most diffusive. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 30–58, 2015

Harmonic diffeomorphisms between domains in the Euclidean 2-sphere

We study the existence or non-existence of harmonic diffeomorphisms between certain domains in the Euclidean 2 -sphere. In particular, we show the existence of harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for maximal graphs in the Lorentzian product \mathbb{M}\times\mathbb{R}_1 , where \mathbb{M} is an arbitrary \mathfrak{n} -dimensional compact Riemannian manifold, \mathfrak{n}\geq 2 . In contrast, we show that there is no harmonic diffeomorphism from the unit complex disc onto the once-punctured sphere and no harmonic diffeomeorphisms from finitely punctured spheres onto circular domains in the Euclidean 2 -sphere.

Maximal Triangle Free Graph and Travelling Salesman Problem

In this paper, a maximal triangle free graph has been generated from the complete graph for by deleting number of edges. In addition, two theorems have been established for it. Finally, an algorithm has been developed under different cases to solve the traveling salesman problem when the weights of the edges are non-repeated of the complete graph for .

Flip distance between triangulations of a planar point set is APX-hard

In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally transforming triangular meshes. We show that, given a point set S in the Euclidean plane and two triangulations T1 and T2 of S, it is an APX-hard problem to minimize the number of edge flips to transform T1 to T2.

Mass angular momentum inequality for axisymmetric vacuum data with small trace

In this paper, we proved the mass angular momentum inequality[16][12][23] for axisymmetric, asymptotically flat, vacuum constraint data sets with small trace. Given an initial data set with small trace, we construct a boost evolution spacetime of the Einstein vacuum equations as [10]. Then a perturbation method is used to solve the maximal surface equation in the spacetime under certain growing condition at infinity. When the initial data set is axisymmetric, we get an axisymmetric maximal graph with the same ADM mass and angular momentum as the given data. The inequality follows from the known results[16][12][23] about the maximal graph.

Multiple-Choice Item Generator for Narrative and Declarative Texts

 Abstract—Multiple Choice Item Generator is a research focused on generating grammatically and semantically correct questions and generating plausible distractors. The system's framework has three main modules: pre-processing, question generation and distractor selection. Pre-processing module implements Hobbs Algorithm for anaphora resolution and direct to indirect speech conversion for handling quoted statements. In generating questions, question overgeneration and ranking framework of Heilman and Smith is used. The distractor selection module uses collocation extraction and Wordnet-based methods such as Lin's semantic similarity measure, hypernyms, hyponyms and maximal bipartite graph matching .