Discrete Spaces: Graphs, Lattices, and Digital Spaces

Abstract

The best way to describe discrete objects is to use graphs. A graph consists of vertices and edges. The vertex usually represents a part of an object, an whole object, or the location of an object; the edge represents a relationship between two vertices. A graph can be defined as \(G=(V,E)\) where V is a set of vertices and E is a set of edges, each of which links two vertices. For a certain geometric object, e.g. a rectangle, one can draw four points on the corners and link them using four edges. The drawback of using edges is that the edge is not a geometric line and it usually does not carry a distance. A geometric space also requires a measurement of distance (the length between two points), called a metric. Therefore, people prefer to use specialized graphs such as triangulated graphs and grid graphs, to represent an object in discrete space. In this chapter, we introduce the discrete spaces made by graphs, lattices, and grid points. We briefly review some of the basic concepts related to discrete objects and discrete spaces.