SIMPLEX ALGORITHM ON A LINEAR GRAPH - A UNIFIED VIEW OF THE EXTREMAL PATH AND CUTSET PROBLEMS ON A GRAPH.

The problem of obtaining the length of a path between a specific pair of nodes on a graph or minmax values of cutset is interesting by itself, but it is a very important problem because it appears as a subproblem for various others. As a solution to this problem, a graphic approach based on labeling method and an algebraic approach represented by linear programming (LP) have been used. In this paper, the above problem is formulated as LP by using Kirchhoff's law represented by fundamental cutset matrix or fundamental loop matrix of a graph, and a new graphic approach having one-to-one correspondence with the simplex method is suggested. It is shown that the problem can be treated in a unified manner. It is further shown that the concepts of base, axis transformation, duality, degeneracy in the simplex method can be related to graph theory.