On semicomplete multipartite digraphs whose king sets are semicomplete digraphs (Q2427514)

Let \(T\) be a semicomplete multipartite digraph with no transmitters and let \(K(r, T)\) denote the set of \(r\)-kings of \(T\). Let \(Q\) be the subdigraph of \(T\) induced by \(K(4, T)\). The paper characterizes all semicomplete digraphs \(D\) which are contained in a semicomplete multipartite digraph whose 4-kings are exactly the vertices of \(D\). It is also shown that \(K(3, Q) = K(3, T)\) for all semicomplete multipartite digraphs \(T\) with no transmitters such that \(Q\) is a semicomplete digraph.