Block digraph of a directed graph

Let D be a connected digraph of order n ; ( n ≥ 3 ) and let B ( D ) = { B 1 , B 2 , … , B N } be a set of blocks of D . The block digraph Q = B ( D ) has vertex set V ( Q ) = B ( D ) and arc set A ( Q ) = B i B j and B i , B j ∈ V ( Q ) , B i , B j have a cut-vertex of D in common and every vertex of B j is reachable from every other vertex of B i We study the properties of B ( D ) and present the characterization of digraphs whose B ( D ) are planar; outerplanar; maximal outerplanar; minimally nonouterplanar; Eulerian; and Hamiltonian.