Symmetry of Extended Representations of Mix Graphs and Sum-Product Graphs

This paper is concerned with two classes of graphs that generalize φ-tolerance chain graphs using the rank tolerance model introduced by Golumbic and Jamison in [3]. In a rank-tolerance representation of a graph, each vertex is assigned two parameters: a rank, which represents the size of that vertex, and a tolerance which represents an allowed extent of conflict with other vertices. Two vertices are adjacent if and only if their joint rank exceeds (or equals) their joint tolerance. In their study, Golumbic and Jamison restrict ranks and tolerances to be positive reals. In certain cases, such as when the tolerance is sum and the ranks is product, it is advantageous to allow all real values. This permits certain transformations which facilitate the study of the representations. We describe the general theory, point out the pivotal place of sum-product graphs and a related class of mix graphs and discuss the special transformations and their usefulness.