A method to find elements of cycles in an incomplete directed graph and its applications—Binary AHP and petri nets

In this paper, we propose an algorithm to find elements of cycles of various lengths in an incomplete directed graph. The proposed algorithm is classified into two cases. We find the cycles of even length and odd length, respectively. In order to find cycles, we need the vertex matrix V corresponding to the directed graph. To find cycles of even length 2m(m = 2, 3, …), we use V and Vm. To find cycles of odd length 2m − 1(m = 2, 3, …), we use V, Vm−1, and Vm. Then we give two kinds of examples to illustrate the usefulness of the proposed algorithm. One is a binary AHP (Analytic Hierarchy Process) and another is a Petri Net. In binary AHP, we apply our algorithm to measure consistency for incomplete comparison case and suggest misjudgments. In Petri Nets, we have T-invariant sets as byproducts of finding even lengths of cycles from an incidence matrix.