Directed Pathos Total Digraph of an Arborescence

For an arborescence A r , a directed pathos total digraph Q = D P T ( A r ) has vertex set V ( Q ) = V ( A r ) ∪ A ( A r ) ∪ P ( A r ) , = where V ( A r ) is the vertex set, A ( A r ) is the arc set, and P ( A r ) is a directed pathos set of A r . The arc set A ( Q ) consists of the following arcs: a b such that a , b ∈ A ( A r ) and the head of a coincides with the tail of b ; u v such that u , v ∈ V ( A r ) and u is adjacent to v ; a u ( u a ) such that a ∈ A ( A r ) and u ∈ V ( A r ) and the head (tail) of a is u ; P a such that a ∈ A ( A r ) and P ∈ P ( A r ) and the arc a lies on the directed path P ; P i P j such that P i , P j ∈ P ( A r ) and it is possible to reach the head of P j from the tail of P i through a common vertex, but it is possible to reach the head of P i from the tail of P j . For this class of digraphs we discuss the planarity; outerplanarity; maximal outerplanarity; minimally nonouterplanarity; and crossing number one properties of these digraphs. The problem of reconstructing an arborescence from its directed pathos total digraph is also presented.