Directed Pathos Total Digraph of an Arborescence

Abstract

For an arborescence Ar, a directed pathos total digraph Q = DPT (Ar) has vertex set V (Q) = V (Ar) ∪ A(Ar) ∪ P (Ar), where V (Ar) is the vertex set, A(Ar) is the arc set, and P (Ar) is a directed pathos set of Ar . The arc set A(Q) consists of the following arcs: ab such that a, b ∈ A(Ar) and the head of a coincides with the tail of b; uv such that u, v ∈ V (Ar) and u is adjacent to v; au (ua) such that a ∈ A(Ar) and u ∈ V (Ar) and the head (tail) of a is u; Pa such that a ∈ A(Ar) and P ∈ P (Ar) and the arc a lies on the directed path P ; PiPj such that Pi, Pj ∈ P (Ar) and it is possible to reach the head of Pj from the tail of Pi through a common vertex, but it is possible to reach the head of Pi from the tail of Pj . 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.