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Degree realization of undirected graphs in reduced
form*1
J. N. Ayoub and I. T. Frisch
Department of Electrical Engineering and Computer Sciences, Electronics Research Laboratory,
University of California Berkeley, California USA
Available online 26 March 2002.
Abstract
Necessary and sufficient conditions are given for a set of non-negative integers to be the degrees of
vertices of an n-vertex undirected graph. The conditions and their proof are based on a novel
decomposition procedure which gives some insight into the structure to be realized, and leads to a
simple and efficient algorithm to construct the graph directly with a minimum number of parallel
branches and a minimum number of self-loops. In the case in which the given set is not realizable, the
algorithm determines the least modification necessary for the resulting set to be realizable.