Line removal algorithms for graphs and their degree lists

An important and basic characterization of a graph is the sequence or list of degrees of that graph. Problems regarding the construction of graphs with specified degrees occur in chemistry and in the design of reliable networks. A list of nonnegative integers is called graphical if there is a graph (called a realization) with the given list as its degree list. The usual algorithms for determining whether a given list is graphical are derived from the effect on a graphical list of the removal of a point from a graph. After reviewing such an algorithm by Havel-Hakimi and its generalization by Wang and Kleitman, we develop a corresponding algorithm based on the removal of a line from a graph. We conclude by reviewing and providing simple proofs of algorithms for a list to be multigraphical due to Hakimi and Butler. The conditions relating a graphical or multigraphical list to the point and line connectivity of their realizations, due to Edmonds, Wang and Kleitman, Boesch and McHugh, and Hakimi, are presented along with new and simple proofs of the multigraph case.