The Antistrong Property for Special Digraph Families

A walk with no arc repeated which begins and ends with forward arcs, in which the arcs alternate between forward and backward arcs, is called forward antidirected trail. A digraph D with order at least three containing a forward antidirected (x, y)-trail for every pair of distinct vertices x, y of D is antistrong. In this paper, we show that the Cartesian product of two antistrong digraphs is antistrong. Moreover, one of the results is that the Lexicographic product of an antistrong digraph and a digraph is antistrong. Finally, the subject is researched to give a necessary and sufficient condition to decide whether a tournament is antistrong.