Given a positive integer $t$, let $P_{ t,2}$ 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 $P_{ t+1,2}$-free digraphs of order $n$, which is denoted by $ex(n, P_{ t+1,2})$. For sufficiently large $n$, we prove that $ex(n, P_{ t+1})=g(n,t)$ when $\lfloor(n-t)/{ 2} \rfloor$ is odd and $ex(n, P_{ t+1,2})\in \{ g(n,t)-1, g(n,t)\}$ when $\lfloor(n-t)/{ 2} \rfloor$ is even, where $g(n,t)=\left\lceil(n+t)/{ 2}\right\rceil \left\lfloor(n-t)/{ 2}\right\rfloor+tn+1$.
Categories: math.CO