Double vertex digraphs of digraphs

Let D be a digraph of order n . The double vertex digraph S 2 ( D ) of D is the digraph whose vertex set consists of all ordered pairs of distinct vertices of V ( D ) such that there is an arc in S 2 ( D ) from ( x , y ) to ( u , v ) if and only if x = u and there is an arc in D from y to v , or y = v and there is an arc in D from x to u . In this paper, we establish some relationships between a digraph and its double vertex digraph.