Graphs and data structures

Abstract

The topology of a triangulation can be described by graph theoretic concepts such that a clear distinction is made between the topological structure and the geometric embedding information. The topological elements of a triangulation are nodes (or vertices), edges and triangles, and the geometric embedding information, which is associated with these elements, is points, curves (or straight-line segments) and surface patches respectively. Likewise, a distinction can be made between topological and geometric operators. By considering triangulations as planar graphs, we can benefit from an extensive theory and a variety of interesting algorithms operating on graphs. In particular, we will see that generalized maps, or G-maps, provide useful algebraic tools to consider triangulations at an abstract level. Common data structures for representing triangulations on computers are outlined and compared in view of storage requirements and efficiency of carrying out topological operations.