Kim A. S. Factor

Kings and Heirs

In 1980, Maurer coined the phrase king when describing any vertex of a tournament that could reach every other vertex in two or fewer steps. A ( 2 , 2 ) -domination graph of a digraph D , d o m 2 , 2 ( D ) , has vertex set V ( D ) , the vertices of D , and edge u v whenever u and v each reach all other vertices of D in two or fewer steps. In this special case of the ( i , j ) -domination graph, we see that Maurers theorem plays an important role in establishing which vertices form the kings that create some of the edges in d o m 2 , 2 ( D ) . But of even more interest is that we are able to use the theorem to determine which other vertices, when paired with a king, form an edge in d o m 2 , 2 ( D ) . These vertices are referred to as heirs. Using kings and heirs, we are able to completely characterize the ( 2 , 2 ) -domination graphs of tournaments.