We develop an experimental algorithm for the exact solving of the maximum independent set problem. The algorithm consecutively finds the maximal independent sets of vertices in an arbitrary undirected graph such that the next such set contains more elements than the preceding one. For this purpose, we use a technique, developed by Ford and Fulkerson for the finite partially ordered sets, in particular, their method for partition of a poset into the minimum number of chains with finding the maximum antichain. In the process of solving, a special digraph is constructed, and a conjecture is formulated concerning properties of such digraph. This allows to offer of the solution algorithm. Its theoretical estimation of running time equals to is $O(n^{8})$, where $n$ is the number of graph vertices. The offered algorithm was tested by a program on random graphs. The testing the confirms correctness of the algorithm.