On the number of nonseparating vertices in strongly connected in-tournaments

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. 1 Terminology and introduction All digraphs mentioned here are finite without loops and multiple arcs. For a digraph D, we denote by V (D) and E(D) the vertex set and arc set of D, respectively. The number |V (D)| is the order of the digraph D. The subdigraph induced by a subset A of V (D) is denoted by D[A]. By D−A we denote the digraph D[V (D)−A]. If A = {x} is a single vertex, then we write D − x instead of D − {x}. If xy ∈ E(D), then y is a positive neighbor of x and x is a negative neighbor of y, and we also say that x dominates y, denoted by x → y. If A and B are two disjoint subdigraphs of a digraph D such that every vertex of A dominates every vertex of B in D, then we say that A dominates B, denoted by A → B. The outset N(x) of a vertex x is the set of positive neighbors of x. More generally, for arbitrary subdigraphs A and B of D, the outset N(A,B) is the set of vertices in B to which there is an arc from a vertex in A. The insets N−(x) and N−(A,B)