
Because of its inherent simplicity, the apparatus of graph theory found extensive use in industrial geometric modeling. One of the wellturned ideas emerged early on was distinguishing between the syntax (topology) and the semantics (geometry) of digital shape representation. In the boundary representation scheme, the topology can be expressed in the form of an acyclic directed simple graph (having no selfloops and parallel edges [Deo, 1974]). The topology graph determines how the primary topological elements (faces, edges, and vertices) are nested. The possibility to have several parent nodes for a single child enables sharing and instancing of the boundary elements.
Let's take a simple box shape in its boundary representation.
The topology graph is the Brep structure itself, so to construct the explicit graph, one has to iterate from the topmost boundary elements (compounds, solids, etc.) down to the atomic entities, i.e., vertices while constructing the corresponding graph nodes and arcs along the way. The following image illustrates a formal topology graph for the box shape.
You can see that even for such simple shapes as boxes, the topology graph is a massive thing. Two options are letting you reduce the complexity of the graph:
The following graph represents a single face selected in the viewer:
The color codes of nodes denote different types of boundary elements.
selected  COMPOUND  COMPSOLID  SOLID  SHELL 
FACE  WIRE  EDGE  VERTEX 
The color codes of arcs denote different topological orientations.
FORWARD  REVERSED 
To specify the level of details, use the corresponding selector in the corresponding dialog box.
The "colorize locations" checkbox allows one to visualize the internal locations of subshapes.
A topology graph is a lowlevel thing. You need to work with it, for example, when implementing Euler operators. The topology graph is also a syntactic descriptor of your CAD model that does not specify any geometry. Therefore, if you do some shape editing, you should first ask yourself if it's a topologyonly operation (like defeaturing of some isolated CAD features). The topologyonly approaches are very efficient and 100% robust if implemented properly.
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