Given a convex polytope P , a set of coordinate functions attached to the vertices of P is barycentric if the functions are non-negative on P and reproduce linear functions sampled at the vertices of P . Warren (1996) describes a construction for rational barycentric coordinate functions of degree n d where d is the dimension of P and n is the number of facets of P . In this paper, we show that...

Given a set S of n points in the plane, we compute in time O(n) the total number of convex polygons whose vertices are a subset of S. We give an O(m n) algorithm for computing the number of convex k-gons with vertices in S, for all values k = 3; : : : ;m; previously known bounds were exponential (O(ndk=2e)). We also compute the number of empty convex polygons (resp., k-gons, k m) with vertices ...

The convex dimension of a graph G = (V, E) is the smallest dimension d for which G admits an injective map f : V −→ R of its vertices into d-space, such that the barycenters of the images of the edges of G are in convex position. The strong convex dimension of G is the smallest d for which G admits a map as above such that the images of the vertices of G are also in convex position. In this pap...

We describe two computational methods for the construction of cubic graphs with given girth. These were used to produce two independent proofs that the (3, 9)-cages, defined as the smallest cubic graphs of girth 9, have 58 vertices. There are exactly 18 such graphs. We also show that cubic graphs of girth 11 must have at least 106 vertices and cubic graphs of girth 13 must have at least 196 ver...

There are numerous results bounding the circumference of certain 3connected graphs. There is no good bound on the size of the largest bond (cocircuit) of a 3-connected graph, however. Oporowski, Oxley and Thomas [11] proved the following result in 1993. For every positive integer k, there is an integer n = f(k) such that every 3-connected graph with at least n vertices contains a Wkor K3,kminor...

‎Recently‎, ‎some techniques such as adding whiskers and attaching graphs to vertices of a given graph‎, ‎have been proposed for constructing a new vertex decomposable graph‎. ‎In this paper‎, ‎we present a new method for constructing vertex decomposable graphs‎. ‎Then we use this construction to generalize the result due to Cook and Nagel‎.

Reconstruction procedure for identifying an unknown polygonal cavity inside a homogeneous, isotropic conductive medium from two boundary measurements is given. The assumptions of the cavity is that there are at most one concave vertex between two convex vertices and the vertices are well separated. The Neumann data of the boundary measurement can change sign at most once and the sets on which o...

Let A1, A2, . . . , An be the vertices of a polygon with unit perimeter, that is ∑n i=1 |AiAi+1| = 1. We derive various tight estimates on the minimum and maximum values of the sum of pairwise distances, and respectively sum of pairwise squared distances among its vertices. In most cases such estimates on these sums in the literature were known only for convex polygons. In the second part, we t...

A cubic graph G is cyclically 5-connected if G is simple, 3–connected, has at least 10 vertices and for every set F of edges of size at most four, at most one component of G\F contains circuits. We prove that if G and H are cyclically 5-connected cubic graphs and H topologically contains G, then either G and H are isomorphic, or (modulo well-described exceptions) there exists a cyclically 5-con...