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  {\bf Abstract}
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Determination of the minimal crossing number of a complete graph is an NP-complete problem that has been attacked in various ways over the years with limited success. Much previous work has focused on conjectured solutions that have been verified only for small values of $n$, where $n$ is the number of vertices in the graph's vertex set. The advent of more powerful computers and the ability to network multiple computers into Beowolf clusters have led to renewed interest in exact solutions to crossing number problems.
 
In this thesis, we introduce a new method for finding the minimal crossing number of complete graphs. Our algorithm lays down new edges independent of each other to create minimal K$_n$ graphs from minimal K$_{n-1}$ graphs, a technique we have entitled Star Analysis. We validate our algorithm on small complete K$_n$ graphs, then demonstrate its efficiency in generating larger minimal K$_n$ graphs up to $n$ = 11. Finally, we extend our process to the generation of minimal K$_{m,n}$ bipartite graphs to document its usefulness in a variety of graph family problems.