On p-competition graphs of loopless Hamiltonian digraphs without symmetric arcs and graph operations

For a digraph $D$, the $p$-competition graph $C_{p}(D)$ of $D$ is satisfying following: $V(C_{p}(D))=V(D)$, for $x,y \in V(C_{p}(D))$, $xy E(C_{p}(D))$ if and only there exist distinct $p$ vertices $v_{1},$ $v_{2},$ $...,$ $v_{p}$ $\in$ $V(D)$ such that $x \rightarrow v_{i}$, $y v_{i}$ $A(D)$ each $i=1,2,$ $p$.