Let T (X∪ Y, A) be a bipartite tournament with partite sets X, Y and arc set A. For any vertex
x∈ X∪ Y, the second out-neighbourhood N++(x) of x is the set of all vertices with distance 2 …

One of the most interesting open problems of a digraph is Seymour's Second Neighborhood
Conjecture (SSNC), which asserts that every digraph D has a vertex v satisfying d++(v)≥ …

For a vertex xx of a digraph, d+(x) d^+(x) (d−(x) d^-(x), respectively) is the number of vertices
at distance 1 from (to, respectively) xx and d++(x) d^++(x) is the number of vertices at …

The Second Neighborhood Conjecture states that every simple digraph has a vertex whose
second out-neighborhood is at least as large as its first out-neighborhood, ie a vertex with …

Abstract Sullivan stated the conjectures:(1) every oriented graph has a vertex x such that
d++(x)≥ d−(x) and (2) every oriented graph has a vertex x such that d++(x)+ d+(x)≥ 2 d−(x) …

Seymour's Second Neighborhood Conjecture asserts that every digraph has a vertex $ v $
whose second out-neighborhood $ N^{++}(v) $ is at least as large as its out-neighborhood …