On the kings and kings-of-kings in semicomplete multipartite digraphs (Q1772420)

A semicomplete multipartite digraph is obtained from a complete multipartite digraph by replacing every edge with an arc or pair of the mutually opposite arcs. A vertex \(x\) of a digraph \(D\) is a \(k\)-king if every vertex of \(D\) is reachable from \(x\) by a path of length at most \(k\). A transmitter is a vertex of zero in-degree. The author proves that if \(D\) is a semicomplete multipartite digraph with no transmitters, then its 4-kings induce a subdigraph with no transmitters. This generalizes a result by Koh and Tan. Further results on 2- and 3-kings are proved. Some problems are posed.