Double vertex digraphs of digraphs

Let DD be a digraph of order nn. The double vertex digraph S2(D)S2(D) of DD is the digraph whose vertex set consists of all ordered pairs of distinct vertices of V(D)V(D) such that there is an arc in S2(D)S2(D) from (x,y)(x,y) to (u,v)(u,v) if and only if x=ux=u and there is an arc in DD from yy to vv, or y=vy=v and there is an arc in DD from xx to uu. In this paper, we establish some relationships between a digraph and its double vertex digraph.