On boolean characterizations of planarity and planar embeddings of graphs

This paper is basically a survey on the Boolean characterizations of planarity and planar embeddings of graphs with some new results. A decomposition of a non-planar graph into a kind of maximal planar subgraphs is also provided.Dedicated to Professor Peter L. Hammer on the occasion of his fiftieth birthdayThis research was partially supported by the National Natural Science Foundation of China.     

On characterizations of classical polynomials

It is well known that the classical families of Jacobi, Laguerre, Hermite, and Bessel polynomials are characterized as eigenvectors of a second order linear differential operator with polynomial coefficients, Rodrigues formula, etc. In this paper we present a unified study of the classical discrete polynomials and q-polynomials of the q-Hahn tableau by using the difference calculus on linear-type lattices. We obtain in a straightforward way several characterization theorems for the classical discrete and q-polynomials of the “q-Hahn tableau”. Finally, a detailed discussion of a characterization by Marcellán et al. is presented.     

Constitutive characterizations of non-ideal constraints in classical mechanics

The theory of two-sided, positional constraints with friction acting on a material point in Classical Mechanics is presented in the geometrical context of the jet-extensions of an affine bundle. This context allows one to study both rigid and non-rigid constraints. We prove that the geometry of the system and the forces acting on the point determine in a natural way the direction of the friction reaction. This allows one to introduce a wide class of constitutive characterizations of constraints with friction, that includes the most common ones.     

Cohomology of classical algebraic groups from the functorial viewpoint

We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study of classical algebraic groups, such as a cohomological stabilization property, the injectivity of external cup products, and the existence of Hopf algebra structures on the (stable) cohomology of a classical algebraic group with coefficients in a Hopf algebra. Our result also opens the way to new explicit cohomology computations. We give an example inspired by recent computations of Djament and Vespa.     

Some combinatorial and geometric characterizations of the finite dual classical generalized hexagons

We characterize the dual of the generalized hexagons naturally associated to the groupsG ₂(q) and³ D ₄(q) by looking at certain configurations, and also by considering intersections of traces. For instance, the dual of a generalized hexagon of finite order (s, t) is associated to the Chevalley groups mentioned above if and only if the intersection of any two tracesx ^y andx ^z, with some additional condition, contains at mostt/s + 1 elements.     

Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. The clique-transversal number and clique-independence number of G are the sizes of a minimum clique-transversal and a maximum clique-independent set of G, respectively. A graph G is clique-perfect if these two numbers are equal for every induced subgraph of G. The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. In this paper, we present a partial result in this direction; that is, we characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to two different subclasses of claw-free graphs.     

Generalized Bosbach states: part I

States have been introduced on commutative and non-commutative algebras of fuzzy logics as functions defined on these algebras with values in [0,1]. Starting from the observation that in the definition of Bosbach states there intervenes the standard MV-algebra structure of [0,1], in this paper we introduce Bosbach states defined on residuated lattices with values in residuated lattices. We are led to two types of generalized Bosbach states, with distinct behaviours. Properties of generalized states are useful for the development of an algebraic theory of probabilistic models for non-commutative fuzzy logics.     

Constructive characterizations of the value-function of a mixed-integer program I

Recent work has provided explicit formulas for the value function of a pure integer program, and for indicator functions for the set of feasible right-hand sides. These formulas are based on linear functions, next-higher integer operations, sums, and maxima. In the present paper, we investigate possible extensions to mixed-integer programs.     

On the planarity of jump graphs

For a graph G of size m1 and edge-induced subgraphs F and H of size k (1km), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uvE(F), wxE(G)−E(F), and H=F−uv+wx. The minimum number of edge jumps required to transform F into H is the k-jump distance from F to H. For a graph G of size m1 and an integer k with 1km, the k-jump graph J_k(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J_k(G) are adjacent if and only if the k-jump distance between the corresponding subgraphs is 1. All connected graphs G for which J₂(G) is planar are determined.     

On the planarity of infinite 2-complexes

This paper is devoted to the problem of characterizing infinite planar polyhedra. We give two topological characterizations of planarity, and two others of proper planarity (embeddings with no accumulation points). We also give a combinatorial characterization of planarity, and proper planarity. In the case of compact polyhedra both results provide a weaker condition than that given by Gross and Rosen in [6].     

On the planarity of Hanoi graphs

Hanoi graphs are the state graphs for Tower of Hanoi problems with three or more pegs. We prove hamiltonicity and present a complete analysis of planarity of these graphs.     

Bending invariants of poly-circles part I: rigidity

For a family of piecewise circular Jordan curves, we show that the curve is determined up to application of Möbius maps, by the pair of bending measured laminations on the boundary of its convex hull in the 3-dimensional hyperbolic space.     

Dynamics of classical quadrupole moment I

It is shown that chaotic behaviour exists in the classical system of the nuclear quadrupole moment under perturbation of the delta function type. The structure of the transformation in the phase space is analysed, and the phenomenology of the transformation discussed by the Poincaré map.     

On the planarity of bicolored digraph grammar systems

D. Wood has asked in [2] whether there exists an inherently non-planar bicolored digraph-grammar language. We shall give below a negative answer to this question.     

Noncommutative plurisubharmonic polynomials part I: Global assumptions

We consider symmetric polynomials, p, in the noncommutative (nc) free variables {x₁,x₂,…,xg}. We define the nc complex hessian of p as the second directional derivative (replacing xT by y)