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doi:10.1016/j.disc.2003.11.049 
Copyright © 2004 Elsevier B.V. All rights reserved.
A smallest irregular oriented graph containing a given diregular one
Received 13 November 2001;
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Abstract
A digraph is called irregular if its vertices have mutually distinct ordered pairs of semi-degrees. Let D be any diregular oriented graph (without loops or 2-dicycles). A smallest irregular oriented graph F, F=F(D), is constructed such that F includes D as an induced subdigraph, the smallest digraph being one with smallest possible order and with smallest possible size. If the digraph D is arcless then V(D) is an independent set of F(D) comprising almost all vertices of F(D) as |V(D)|→∞. The number of irregular oriented graphs is proved to be superexponential in their order. We could not show that almost all oriented graphs are/are not irregular.
Author Keywords: Irregularization; Diregular digraph; Oriented graph; Almost all vertices; Superexponential cardinality
