A robust method for searching the smallest set of smallest rings with a path-included distance matrix

The perception of rings in graphs is widely used in many fields of science and engineering. Algorithms developed in the chemistry community, called smallest set of smallest rings (SSSR), applicable only for simple graphs or chemical structures. In contrast, algorithms developed by the computer science community, called minimum cycle basis (MCB) are identical to SSSR yet exhibit greater robustness. MCB-based algorithms can correctly reveal all rings in any complex graph. However, they are slow when applied to large complex graphs due to the inherent limitations of the algorithms used. Here, we suggest a heuristic method called RP-Path. This method is a robust, simple, and fast search method with O(n(3)) runtime algorithm that correctly identifies the SSSR of all of the test case of complex graphs by using approach different from the MCB-based method. Both the robustness and improvement in speed are achieved by using a path-included distance matrix and describing the characteristic features of rings in the matrix. This method is accurate and faster than any other methods and may find many application in various fields of science and engineering that use complicated graphs with thousands of nodes.