Extremal digraphs containing at most t paths of length 2 with the same endpoints

Given a positive integers t, let Pₓ, ₂ be the digraph consisting of t directed paths of length 2 with the same initial and terminal vertices. In this paper, we study the maximum size of Pₓ+₁, ₂-free digraphs of order n, which is denoted by ex (n, Pₓ+₁). For sufficiently large n, we prove that ex (n, Pₓ+₁) =g (n, t) when (n-t) /2 is odd and ex (n, Pₓ+₁) \g (n, t) -1, g (n, t) \ when (n-t) /2 is even, where g (n, t) = (n+t) /2 (n-t) /2+tn+1.