The study of cages focuses on finding (k, g)-graphs of minimal order. This dissertation generalizes the problem of finding cages to the determination of graphs with specified excess, thereby broadening the significance of the results. The (k, g, ε)-graph problem seeks to determine the existence or nonexistence of k-regular graphs with girth g and excess ε = n(G)−M(k, g) (where M(k, g) represents the Moore bound for cage graphs). Motivated by heuristic methods used to determine properties ...
