Geometric graph theory

Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are allowed to be arbitrary continuous curves connecting the vertices; thus, it can be described as "the theory of geometric and topological graphs" (Pach 2013). Geometric graphs are also known as spatial networks. A planar straight-line graph is a graph in which the vertices are embedded as points in the Euclidean plane, and the edges are embedded as non-crossing line segments. Fáry's theorem states that any planar graph may be represented as a planar straight line graph. A triangulation is a planar straight line graph to which no more edges may be added, so called because every face is necessarily a triangle; a special case of this is the Delaunay triangulation, a graph defined from a set of points in the plane by connecting two points with an edge whenever there exists a circle containing only those two points. The 1-skeleton of a polyhedron or polytope is the set of vertices and edges of said polyhedron or polytope. The skeleton of any convex polyhedron is a planar graph, and the skeleton of any k-dimensional convex polytope is a k-connected graph. Conversely, Steinitz's theorem states that any 3-connected planar graph is the skeleton of a convex polyhedron; for this reason, this class of graphs is also known as the polyhedral graphs. A Euclidean graph is a graph in which the vertices represent points in the plane, and the edges are assigned lengths equal to the Euclidean distance between those points. The Euclidean minimum spanning tree is the minimum spanning tree of a Euclidean complete graph. It is also possible to define graphs by conditions on the distances; in particular, a unit distance graph is formed by connecting pairs of points that are a unit distance apart in the plane.

A Note on Lenses in Arrangements of Pairwise Intersecting Circles in the Plane

Rom Pinchasi

Let F be a family of n pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by F is at most 2n - 2. This bound is tight. Furthermore, if no two circles in F touch, then the geometr ...

Electronic Journal Of Combinatorics2024

A Tutorial-Cum-Survey on Percolation Theory With Applications in Large-Scale Wireless Networks

Ainur Zhaikhan

Connectivity is an important key performance indicator and a focal point of research in large-scale wireless networks. Due to path-loss attenuation of electromagnetic waves, direct wireless connectivity is limited to proximate devices. Nevertheless, connec ...

Ieee-Inst Electrical Electronics Engineers Inc2024

A Note on Lenses in Arrangements of Pairwise Intersecting Circles in the Plane

A Tutorial-Cum-Survey on Percolation Theory With Applications in Large-Scale Wireless Networks

ON THE INDEPENDENCE NUMBER OF INTERSECTION GRAPHS OF AXIS-PARALLEL SEGMENTS

ON THE INDEPENDENCE NUMBER OF INTERSECTION GRAPHS OF AXIS-PARALLEL SEGMENTS

A Tutorial-Cum-Survey on Percolation Theory With Applications in Large-Scale Wireless Networks

A Note on Lenses in Arrangements of Pairwise Intersecting Circles in the Plane