Projective geometry of polygons and discrete 4-vertex and 6-vertex theorems

The Four-Vertex Theorem, The Evolute, and The Decomposition of Polygons

The Four-Vertex Theorem has been of interest ever since a discrete version appeared in 1813 due to Cauchy. Up until now, there have been many different versions of this theorem, both for discrete cases and smooth cases. In 2004 [16], an approach relating the discrete Four-Vertex Theorem to the evolute was published, and here we will give an overview of this paper. We then will define the notion...

Bounds for the Regularity of Edge Ideal of Vertex Decomposable and Shellable Graphs

Higher pentagram maps, weighted directed networks, and cluster dynamics

The pentagram map was introduced by R. Schwartz about 20 years ago [25]. Recently, it has attracted a considerable attention: see [11, 16, 17, 20, 21, 22, 26, 27, 28, 29, 30] for various aspects of the pentagram map and related topics. On plane polygons, the pentagram map acts by drawing the diagonals that connect second-nearest vertices and forming a new polygon whose vertices are their consec...

Wiener, Szeged and vertex PI indices of regular tessellations

A lot of research and various techniques have been devoted for finding the topological descriptor Wiener index, but most of them deal with only particular cases. There exist three regular plane tessellations, composed of the same kind of regular polygons namely triangular, square, and hexagonal. Using edge congestion-sum problem, we devise a method to compute the Wiener index and demonstrate th...

Guarding curvilinear art galleries with vertex

10 One of the earliest and most well known problems in computational geometry is 11 the so-called art gallery problem. The goal is to compute the minimum number of 12 guards needed to cover the interior of any polygon with n vertices; the guards are 13 to be placed on the vertices of the polygon. We consider the problem of guarding an 14 art gallery which is modeled as a polygon with curvilinea...