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Weak king embeddings of oriented graphs

The concept of weak kings in oriented graphs, which extends the notion of kings in tournaments, was introduced by Pirzada and Shah in 2008. In this article we show that almost every oriented graph D is an induced subgraph of both an all-weak-kings oriented graph D′ and an oriented graph having only the vertices of D as its weak kings. We also obtain lower bounds on the sizes of these supergraphs, where the one for allweak-kings oriented graphs D′ can be achieved for every D. These are analogues to the results proved by Reid (1982) for kings in tournaments.