Deniz Sarioz

Convex Obstacle Numbers of Outerplanar Graphs and Bipartite Permutation Graphs

The disjoint convex obstacle number of a graph G is the smallest number h such that there is a set of h pairwise disjoint convex polygons (obstacles) and a set of n points in the plane [corresponding to V (G))]so that a vertex pair uv is an edge if and only if the corresponding segment \(\overline{uv}\) does not meet any obstacle.

Graphs with large obstacle numbers

Motivated by questions in computer vision and sensor networks, Alpert et al. [3] introduced the following definitions. Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of connected obstacles such that two vertices of G are joined by an edge if an only if the corresponding points can be connected by a segment which avoids all obstacles. The obstacle number of G is the minimum number of obstacles in an obstacle representation of G. It was shown in [3] that there exist graphs of n vertices with obstacle number at least Ω(√logn). We use extremal graph theoretic tools to show that (1) there exist graphs of n vertices with obstacle number at least Ω(n/log2 n), and (2) the total number of graphs on n vertices with bounded obstacle number is at most 2o(n2). Better results are proved if we are allowed to use only convex obstacles or polygonal obstacles with a small number of sides.

History trees as descriptors of macromolecular structures

High-level structural information about macromolecules is now being organized into databases. One of the common ways of storing information in such databases is in the form of three-dimensional (3D) electron microscopic (EM) maps, which are 3D arrays of real numbers obtained by a reconstruction algorithm from EM projection data. We propose and demonstrate a method of automatically constructing, from any 3D EM map, a topological descriptor (which we call a history tree) that is amenable to automatic comparison.

More is more: The benefits of dense sensor deployment

An ad-hoc sensor network is composed of sensing devices which can measure or detect features of their environment, communicate with one other and possibly with other devices that perform data fusion. One of the problems motivated by ad-hoc sensor networks is to position sensors in order to maximize coverage, or equivalently to minimize the number of sensors required to cover a given area.