Gromov hyperbolicity of periodic planar graphs

The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it.The main result in this paper is a very simple characterization of the hyperbolicity of a large class of periodic planar graphs.     

Bounds on Gromov hyperbolicity constant in graphs

If X is a geodesic metric space and x ₁,x ₂,x ₃ ∈ X, a geodesic triangle T = {x ₁,x ₂,x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e. $\delta(X)=\inf\{\delta\ge 0: \, X \, \text{ is $\delta(X)=\inf\{\delta\ge 0: \, X \, \text{ is In this paper we relate the hyperbolicity constant of a graph with some known parameters of the graph, as its independence number, its maximum and minimum degree and its domination number. Furthermore, we compute explicitly the hyperbolicity constant of some class of product graphs.     

On hyperbolicity cones associated with elementary symmetric polynomials

Elementary symmetric polynomials can be thought of as derivative polynomials of . Their associated hyperbolicity cones give a natural sequence of relaxations for . We establish a recursive structure for these cones, namely, that the coordinate projections of these cones are themselves hyperbolicity cones associated with elementary symmetric polynomials. As a consequence of this recursion, we give an alternative characterization of these cones, and give an algebraic characterization for one particular dual cone associated with together with its self-concordant barrier functional.     

Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces

In this paper we obtain the equivalence of the Gromov hyperbolicity between an extensive class of complete Riemannian surfaces with pinched negative curvature and certain kind of simple graphs, whose edges have length 1, constructed following an easy triangular design of geodesics in the surface.     

Twists and Gromov hyperbolicity of riemann surfaces

The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann surfaces (with their Poincaré metrics). We prove that quasiconformal maps between Riemann surfaces preserve hyperbolicity; however, we also show that arbitrary twists along simple closed geodesics do not preserve it, in general.     

On conditional edge-connectivity of graphs

1. IntroductionIn this paper, a graph G ~ (V,E) always means a simple graph (without loops andmultiple edges) with the vertex-set V and the edge-set E. We follow [1] for graph-theoreticalterllilnology and notation not defined here.It is well known that when the underlying topology of a computer interconnectionnetwork is modeled by a graph G, the edge-connectivity A(G) of G is an important measurefor fault-tolerance of the network. However, it has many deficiencies (see [2]). MotiVatedby t…     

Gromov hyperbolicity of Denjoy Domains

In this paper we characterize the Gromov hyperbolicity of the double of a metric space. This result allows to give a characterization of the hyperbolic Denjoy domains, in terms of the distance to of the points in some geodesics. In the particular case of trains (a kind of Riemann surfaces which includes the flute surfaces), we obtain more explicit criteria which depend just on the lengths of what we have called fundamental geodesics. Research partially supported by three grants from M.E.C. (MTM 2006-11976, MTM 2006-13000-C03-02 and MTM 2004-21420-E), Spain.     

Geometric tools to determine the hyperbolicity of limit cycles

In this paper we present a new method to study limit cycles' hyperbolicity. The main tool is the function ν=([V,W]∧V)/(V∧W), where V is the vector field under investigation and W a transversal one. Our approach gives a high degree of freedom for choosing operators to study the stability. It is related to the divergence test, but provides more information on the system's dynamics. We extend some previous results on hyperbolicity and apply our results to get limit cycles' uniqueness. Liénard systems and conservative + dissipative systems are considered among the applications.     

Characterization of Gromov hyperbolic short graphs

To decide when a graph is Gromov hyperbolic is,in general,a very hard problem.In this paper,we solve this problem for the set of short graphs(in an informal way,a graph G is r-short if the shortcuts in the cycles of G have length less than r):an r-short graph G is hyperbolic if and only if S9r(G)is finite,where SR(G):=sup{L(C):C is an R-isometric cycle in G}and we say that a cycle C is R-isometric if dC(x,y)≤dG(x,y)+R for every x,y∈C.     

On the diameter of Kronecker graphs

We prove that a.a.s. as soon as a Kronecker graph becomes connected it has a finite diameter.     

On the connectivity and superconnected graphs with small diameter

In this paper, first we prove that any graph G is 2-connected if diam(G)≤g−1 for even girth g, and for odd girth g and maximum degree Δ≤2δ−1 where δ is the minimum degree. Moreover, we prove that any graph G of diameter diam(G)≤g−2 satisfies that (i) G is 5-connected for even girth g and Δ≤2δ−5, and (ii) G is super-κ for odd girth g and Δ≤3δ/2−1.     

Gromov hyperbolicity in Cartesian product graphs

If X is a geodesic metric space and x ₁, x ₂, x ₃ ∈ X, a geodesic triangle T = {x ₁, x ₂, x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e. δ(X) = inf{δ ≥ 0: X is δ-hyperbolic}. In this paper we characterize the product graphs G ₁ × G ₂ which are hyperbolic, in terms of G ₁ and G ₂: the product graph G ₁ × G ₂ is hyperbolic if and only if G ₁ is hyperbolic and G ₂ is bounded or G ₂ is hyperbolic and G ₁ is bounded. We also prove some sharp relations between the hyperbolicity constant of G ₁ × G ₂, δ(G ₁), δ(G ₂) and the diameters of G ₁ and G ₂ (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the precise value of the hyperbolicity constant for many product graphs.     

Minimum node covers and 2-bicritical graphs

The problem of finding a minimum cardinality set of nodes in a graph which meet every edge is of considerable theoretical as well as practical interest. Because of the difficulty of this problem, a linear relaxation of an integer programming model is sometimes used as a heuristic. In fact Nemhauser and Trotter showed that any variables which receive integer values in an optimal solution to the relaxation can retain the same values in an optimal solution to the integer program. We define 2-bicritical graphs and give several characterizations of them. One characterization is that they are precisely the graphs for which an optimal solution to the linear relaxation will have no integer valued variables. Then we show that almost all graphs are 2-bicritical and hence the linear relaxation almost never helps for large random graphs.This research was supported in part by the National Research Council of Canada.     

On perfect graphs and polyhedra with (0, 1)-valued extreme points

LetC andA be (0, 1)-valued matrices with no zero columns. Fulkerson has shown that the extreme points of {x: Cx 1,x 0} are given by the rows ofA and their projections and the extreme points of {x: Ax 1,x 0} are given by the rows ofC and their projections if and only if the maximal rows ofC andA are the incidence vectors of maximal cliques and anticliques, respectively, of a perfect graph. This theorem is discussed and a new proof is given for the only if implication.Research partially supported by grant ENG 76-09936 from the National Science Foundation to Cornell University and by Sonderforschungsbereich 21 (DFG), Institut für Operations Research, Universität Bonn.     

On Homotopy of connected graphs having the same degree function

Let denote the set of simple graphs having the degree function . It is known that any two graphs from are homotopic in the sense that one can be obtained from the other by a series of simple two-edge switches so that every intermediate graph is in . Let denote the set of simple graphs having at most k components. A natural question arises whether it is possible to extend the above homotopy result on the class , and in particular, whether it is possible to obtain a connected simple graph from another connected simple graph of the same degree function by a series of two edge switches preserving connectivity. In this paper we give an answer to this question. We also consider two-edge switches of a special type and single out some classes of graphs for which these special operations provide the corresponding homotopy between its members.     

On total domination vertex critical graphs of high connectivity

A graph is called γ-critical if the removal of any vertex from the graph decreases the domination number, while a graph with no isolated vertex is γ_t-critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A γ_t-critical graph that has total domination number k, is called k-γ_t-critical. In this paper, we introduce a class of k-γ_t-critical graphs of high connectivity for each integer k≥3. In particular, we provide a partial answer to the question “Which graphs are γ-critical and γ_t-critical or one but not the other?” posed in a recent work [W. Goddard, T.W. Haynes, M.A. Henning, L.C. van der Merwe, The diameter of total domination vertex critical graphs, Discrete Math. 286 (2004) 255-261].     

Equimatchable claw-free graphs

A graph is equimatchable if all of its maximal matchings have the same size. A graph is claw-free if it does not have a claw as an induced subgraph. In this paper, we provide the first characterization of claw-free equimatchable graphs by identifying the equimatchable claw-free graph families. This characterization implies an efficient recognition algorithm.