Simplex Adjacency Graphs in Linear Optimization

Simplex Adjacency graphs represent the possible Simplex pivot operations (the edges) between pairs of feasible bases (the nodes) of linear optimization models. These graphs are mainly studied so far in the context of degeneracy. The more general point of view in this paper leads to a number of new results, mainly concerning the connectivity and the feasibility-optimality duality in these graphs. Among others, we present a very short proof of a result of P. Zörnig and T. Gall (1996) on the connectivity of subgraphs corresponding to optimal vertices, and we answer H.-J. Kruse’s (1993) question on the connectivity of graphs related to the negative pivots in optimal vertices.