Minimal path decomposition of complete bipartite graphs

This paper deals with the subject of minimal path decomposition of complete bipartite graphs. A path decomposition of a graph is a decomposition of it into simple paths such that every edge appears in exactly one path. If the number of paths is the minimum possible, the path decomposition is called minimal. Algorithms that derive such decompositions are presented, along with their proof of correctness, for the three out of the four possible cases of a complete bipartite graph.