The decomposition of linkages into Assur graphs (Assur groups) was developed by Leonid Assur in 1914 - to decompose a linkage into fundamental minimal components for the analysis and synthesis of the linkages. In the paper, some new results and new methods are introduced for solving problems in mechanisms, bringing in methods from the rigidity theory community. Using these techniques, an investigation of Assur graphs and the decomposition of linkages has reworked and extended the decomposition using the well developed mathematical concepts from theory of rigidity and directed graphs. We recall some vocabulary and provide an efficient algorithm for decomposing 2-dimensional linkages into Assur components using strongly connected decompositions of graphs and a fast combinatorial Pebble Game Algorithm, which has been recently used in the study of rigidity and flexibility of structures and in fast analysis of large biomolecular structures such as proteins. Working on a one degree of freedom mechanism, we apply our algorithm to give the Assur decomposition. The Pebble Game Algorithm on such a mechanism is presented, along with an overview of the key properties and advantages of this elegant algorithm. We show how the pebble game algorithm can be used in the analysis and synthesis of linkages to mechanical engineering community. Core techniques and algorithms easily generalize to 3-dimensional structures, and can be further adapted to entire suite of other (body-bar) types of kinematic structures.

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