Ottilia Fülöp

Star graph representations of chiral objects in graph theory

Planar chirality of objects is a problem with important applications in many fields of natural sciences, especially in chemistry and pharmacology. The analysis of chirality properties can be studied using n-polyominoes and planar graphs. In this paper we show that graph representations of chiral objects can be star-graphs.

Three-dimensional chiral objects and their star graph representations

Planar chirality properties can be analysed using n-polyominoes and graphs. In this paper we study graph representations of three-dimensional chiral objects and discuss the generalization of planar case. We show that graph representations of three-dimensional chiral objects can be star graphs.

Graph Theoretical and Statistical Analysis of the Impact of Soai Reaction on Natural Sciences

Asymmetric autocatalysis (Soai reaction) was reported 20 years ago. Since then, this discovery has had an exceptional influence on several branches of natural sciences. In the present paper citations to this milestone article—556 publications of 908 scientists from Google Scholar’s database—are analyzed by graph theoretical and statistical tools. The results enable localization of the most active research groups, their priorities and the identification of the distribution of scientific disciplines influenced by the Soai reaction.

Graph Theoretical Tools in Two- and Three-Dimensional Chirality Problems

Abstract Two-dimensional chirality problems in mathematics can be studied by so-called lattice animals or 2D animals. Generalizing this idea for 3D problems we consider in the three-dimensional space R3 a Cartesian grid of the first octant consisting of n3 small unit cubic cells and define there solid animals or 3D animals. The simplest approach is to say that a 2D (or 3D) animal is chiral or achiral. One could assume that there is no other possibility. On the other hand a few scientists recognized that it is possible to measure the degree of chirality. For example F. Harary, R. W. Robinson, P. G. Mezey, A. I. Kitaigorodski, K. Mislow, J. Siegel, G. Gilat, D. Avnir, and A. Y. Meyer have made important contributions to this field. In this paper we provide a solution for quantification of the problem of chirality for 2D and 3D objects using graphs.Two-dimensional chirality problems in mathematics can be studied by so-called lattice animals or 2D animals. Generalizing this idea for 3D problems we consider in the three-dimensional space R3 a Cartesian grid of the first octant consisting of n3 small unit cubic cells and define there solid animals or 3D animals. The simplest approach is to say that a 2D (or 3D) animal is chiral or achiral. One could assume that there is no other possibility. On the other hand a few scientists recognized that it is possible to measure the degree of chirality. For example F. Harary, R. W. Robinson, P. G. Mezey, A. I. Kitaigorodski, K. Mislow, J. Siegel, G. Gilat, D. Avnir, and A. Y. Meyer have made important contributions to this field. In this paper we provide a solution for quantification of the problem of chirality for 2D and 3D objects using graphs.

Sparse graph certificates for mixed connectivity

We give a short proof for the Mixed Connectivity Certificate Theorem of Even, Itkis and Rajsbaum and provide an upper bound on the edge number of a certificate of local T-mixed connectivity up to k.

Fast Algorithms for Even/Odd Minimum Cuts and Generalizations

We give algorithms for the directed minimum odd or even cut problem and certain generalizations. Our algorithms improve on the previous best ones of Goemans and Ramakrishnan by a factor of O(n) (here n is the size of the ground vertex set). Our improvements apply among others to the minimum directed Todd or T-even cut and to the directed minimum Steiner cut problems. The (slightly more general) result of Goemans and Ramakrishnan shows that a collection of minimal minimizers of a submodular function (i.e. minimum cuts) contains the odd minimizers. In contrast our algorithm selects an n-times smaller class of not necessarily minimal minimizers and out of these sets we construct the odd minimizer. If M(n,m) denotes the time of a u-v minimum cut computation in a directed graph with n vertices and m edges, then we may find a directed minimum - odd or T-odd cut with V (or T) even in O(n2m + n ċ M(n,m)) time; - even or T-even cut in O(n3m + n2 ċ M(n, m)) time. The key of our construction is a so-called parity uncrossing step that, given an arbitrary set system with odd intersection, finds an odd set with value not more than the maximum of the initial system.