  The paper presents an algorithm for minimum vertex cover problem, which is an
NP-Complete problem. The algorithm computes a minimum vertex cover of each
input simple graph. Tested by the attached MATLAB programs, Stage 1 of the
algorithm is applicable to, i.e., yields a proved minimum vertex cover for,
about 99.99% of the tested 610,000 graphs of order 16 and 99.67% of the tested
1,200 graphs of order 32, and Stage 2 of the algorithm is applicable to all of
the above tested graphs. All of the tested graphs are randomly generated graphs
of random "edge density" or in other words, random probability of each edge. It
is proved that Stage 1 and Stage 2 of the algorithm run in $O(n^{5+logn})$ and
$O(n^{3(5+logn)/2})$ time respectively, where $n$ is the order of input graph.
Because there is no theoretical proof yet that Stage 2 is applicable to all
graphs, further stages of the algorithm are proposed, which are in a general
form that is consistent with Stages 1 and 2.
