On the number of nonseparating vertices in strongly connected in-tournaments
Dirk Meierling · 2009
A digraph without loops, multiple arcs and directed cycles of length two is called an in-tournament if the set of in-neighbors of every vertex induces a tournament. A local tournament is an in-tournament such that the set of out-neighbors of every vertex induces a tournament as well. Let p ≥ 2 be an integer and let T be a strongly connected tournament such that every vertex has at least p positive neighbors and at least p negative neighbors. In 2006, Kotani showed that T has at least k vertices x1,x2,...,xk, where k =min{|V (D)|, 4p − 2}, such that T − xi (i =1, 2,...,k) is strongly connected. One year later, Meierling and Volkmann proved that the same proposition is valid for the class of local tournaments. In this paper we shall generalize the result to the class of in-tournaments, thereby generalizing Kotani’s as well as Meierling’s and Volkmann’s results.