In order to solve the reachability problem in Petri nets, reachability tree is one of useful method. To constructing the reachability tree, firing count vector will give available information. Firing count vector can be get as a solution of the equation to incidence matrix. In ordinary cases, an incidence matrix does not have full rank which is necessary condition to compute the inverse matrix for the solutions of the matrix equation. A method to make full rank matrix from an incidence matrix is to add some counter-places into the incidence matrix. But, in this method, cycle structures and/or parallel structures must be identified and total cycle numbers and parallel numbers must be counted up. Up to the present, there is no way how to count cycle and parallel numbers in a given Petri net. In this paper, we give a method to add counter-places by algebrical way without identifying the cycle structures and parallel structures in a Petri net. Let r be the rank of the column vectors of an m×n incidence matrix, we prove that the number which is the sum of cycles and parallels must be n-r. Further, we propose a method to identify the cycle structures and/or parallel structures by the solutions of the homogeneous system of transition invariant. We, also, prove that the counter-places added in the method presented in this paper, must be added one by one into the cycles and parallels, respectively.
