A fast algorithm for kinematic chain isomorphism identification based on dividing and matching vertices

Kinematic chain isomorphism identification is a crucial issue in mechanism topology and an important application of Graph Isomorphism to mechanisms. In this paper, a kinematic chain is uniquely represented by a graph, and a fast deterministic algorithm called the Dividing and Matching Algorithm (DMA) is proposed. First, the vertices of each graph are divided by the degree. Then, vertex connection properties in a sub-graph and between sub-graphs are explored. Accordingly the expanded square degree and the correlation degree are proposed, based on which, the Dividing Vertex Algorithm (DVA) is developed to divide vertices into sets. Moreover, it is proved that only the vertices from the corresponding sets between two graphs are possible to be bijective or matched, which avoids exhaustive search. Eventually, a backtracking procedure is employed to match the vertices between corresponding sets by calling up DVA repeatedly. DMA detects whether the adjacency matrices of two graphs can be adjusted to be equivalent by changing the orders of vertices. Justifications for the reliability of each part of DMA are provided. The experiments and comparisons with existing algorithms show the effectiveness and efficiency of DMA.