It is well known that every planar kinematical linkage can be decomposed into basic topological structures referred as Assur Groups. A new reformulation of Assur Group concept into the terminology of rigidity theory, as Assur Graphs, has yielded the development of new theorems and methods. The paper reports on an algorithm for systematic construction of Assur Graph classes, termed fundamental Assur Graphs. From each fundamental Assur Graph it is possible to derive an infinite set of different Assur Graphs. This mapping algorithm is proved to be complete and sound, i.e., all the Assur Graphs appear in the map and each graph in the map is an Assur Graph. Once we possess the mapping of all the Assur Graphs, all valid kinematical linkage topologies can be constructed through various Assur Graph compositions.

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