Mapping Of Vertices Research Articles

A flexible triangulation method to describe the solvent-accessible surface of biopolymers.

A relatively simple protein solvent-accessible surface triangulation method for continuum electrostatics applications employing the boundary element method is presented. First, the protein is placed onto a three-dimensional lattice with a specified lattice spacing. To each lattice point, a box is assigned. Boxes located in the solvent region and in the interior of the protein are removed from the set. Improper connections between boxes and the possible existence of cavities in the interior of the protein which would destroy the proper connectivity of the triangulated surface are taken care of. The remaining set of boxes define the outer contour of the protein. Each free face exposed to the solvent of the remaining set of boxes is triangulated after the surface defined by the free faces has been smoothed to follow the shape of the macromolecule more accurately. The final step consists of a mapping of triangle vertices onto a set of surface points which define the solvent-accessible surface. Normal vectors at triangle vertices are obtained also from the free faces which define the orientation of the surface. The algorithm was tested for six molecules including four proteins; a dipeptide, a helical peptide consisting of 25 residues, calbindin, lysozyme, calmodulin and cutinase. For each molecule, total areas have been calculated and compared with the result computed from a dotted solvent-accessible surface. Since the boundary element method requires a low number of vertices and triangles to reduce the number of unknowns for reasons of efficiency, the number of triangles should not be too high. Nevertheless, credible results are obtained for the total area with relative errors not exceeding 12% for a large lattice spacing (0.30 nm) while close to zero for a smaller lattice spacing (down to 0.16 nm). The output of the triangulation computer program (written in C++) is rather simple so that it can be easily converted to any format acceptable for any molecular graphics programs.

Multiplicativity of acyclic digraphs

A homomorphism of a digraph to another digraph is an edge preserving vertex mapping. A digraph W is said to be multiplicative if the set of digraphs which cannot be homomorphically mapped to W is closed under categorical product. We discuss the necessary conditions for a digraph to be multiplicative. Our main result is that almost all acyclic digraphs which have a Hamiltonian path are nonmultiplicative. We conjecture that almost all digraphs are nonmultiplicative.

Multiplicativity of acyclic local tournaments

A homomorphism of a digraph to another digraph is an edge-preserving vertex mapping. A digraphH is said to be multiplicative if the set of digraphs which do not admit a homomorphism toH is closed under categorical product. In this paper we discuss the multiplicativity of acyclic Hamiltonian digraphs, i.e., acyclic digraphs which contains a Hamiltonian path. As a consequence, we give a complete characterization of acyclic local tournaments with respect to multiplicativity.

UniquelyH-colorable graphs with large girth

Suppose G and H are graphs. We say G is H-colorable if there is a homomorphism (edge-preserving vertex mapping) from G to H. We say a graph G is uniquely H-colorable if there is an onto homomorphism c from G to H, and any other homomorphism from G to H is the composition σ ??? c of c with an automorphism σ of H. In case H is the complete graph Kn, the notion of uniquely H-colorable coincides with that of uniquely n-colorable. It was proved by B. Bollobás and N. Sauer that for any integer n there are graphs of arbitrary large girth which are uniquely n-colorable. We generalize this result and prove that for any graph H which is a core (i.e., H admits no homomorphisms to any of its proper subgraphs), and for any integer g, there is a graph which is uniquely H-colorable and has girth at least g. We then use this generalization to answer a question concerning the star-chromatic number of a graph. A graph G is said to be (k,d)-colorable if there is a coloring c of the vertices of G with k colors {0,1,…,k − 1} such that d ≤ |c(x)−c(y)| ≤ k − d for every edge (x,y) of G. The star-chromatic number of G is the infimum of the ratio k/d such that G is (k,d)-colorable. We shall show that for any rational k/d ≥ 2, there are graphs G of arbitrary large girth with star-chromatic number χ*(G) = k/d. In particular, for any integer n, there are graphs G of arbitrary large girth with χ*(G) = χ(G)=n. © 1996 John Wiley & Sons, Inc.

On homomorphisms to acyclic local tournaments

AbstractA homomorphism of a digraph to another digraph is an edgepreserving vertex mapping. A local tournament is a digraph in which the inset as well as the outset of each vertex induces a tournament. Thus acyclic local tournaments generalize both directed paths and transitive tournaments. In both these cases there is a simple characterization of homomorphic preimages. Namely, if H is a directed path, or a transitive tournament, then G admits a homomorphism to H if and only if each oriented path which admits a homomorphism to G also admits a homomorphism to H. We prove that this result holds for all acyclic local tournaments. © 1995 John Wiley & Sons, Inc.

Minimum Integral Drawings of the Platonic Graphs

(1994). Minimum Integral Drawings of the Platonic Graphs. Mathematics Magazine: Vol. 67, No. 5, pp. 355-358.

Efficient algorithms for dilated mappings of binary trees

We address the problem of finding a 1-1 mapping of vertices of a binary tree onto those of a target binary tree such that the son of one binary tree node is mapped onto a descendent of the image of that node in the other binary tree. There are two natural measures of the cost of this mapping, namely the dilation-cost, i.e., one less than the maximum distance in the target binary tree between the images of vertices that are adjacent in the other graph. The other measure, i.e., the expansion-cost, is defined as the number of extra nodes/edges to be added into the target binary tree in order to ensure a 1-1 mapping. In this paper, we describe an efficient algorithm which can be used to find a mapping of one binary tree onto a target binary tree. We show that it is possible to minimize one cost of the mapping at the expense of increase in the other.

Iso-orientation areas in the foveal cone mosaic

The quality of the foveal cone mosaic in human and primate retinas is a basic parameter of spatial vision function. The present study uses digital-texture analysis procedures to analyze the crystalline order of inner segment sections containing the rod-free portions of foveal cone mosaics. Definition of the cone cross-sectional centers made possible by adequate preprocessing allows precise mapping of lattice vertices and differentiation of hexagonal positions by procedures for direct neighbor recognition. In a further step, the existence of subunits within the hexagonal areas is revealed by the determination of axial orientation. The lattice of the subunits is characterized by similar orientation and high positional correlation of its hexagonal units. The axial orientation of the areas differs from that of neighboring subunits by angular shifts of 10-15 deg and linear series of nonhexagonal irregularities demarcate the borders. Although larger patches with continuous hexagonal order occur in the surrounding rod-free regions, elevated degrees of disorder (30%) are found within the foveolar center (ca. 300 cones). Analysis of a mosaic showing labeled B cones (Szél et al., 1988) demonstrates that lattice disorder is in part associated with the blue cone subpopulation. The foveal mosaic from a glaucomatuous eye reveals severe lattice degradation throughout the rod-free zone, presumably due to extensive receptor loss. The low-frequency superstructure results in local sets of sampling grids (5'-8') with differing orientational bias. Besides a horizontal/vertical difference of mosaic compression (ca. 1:1.15), the present analysis gives no hints for the existence of systematic meridional anisotropies at the receptor mosaic level. The study reveals a discontinuous organization of the foveal mosaic and points to possible sources for the induction and location of lattice disorder.

The asymptotic number of tree-rooted maps on a surface

Let S be a surface. We asymptotically enumerate two classes of n-edged maps on S as N → ∞: tree-rooted and tree-rooted smooth. These results are based on a system of equations enumerating single vertex maps and on a relation found by Walsh and Lehman for the case of orientable surfaces between tree-rooted and single vertex maps.