On the Planarity of a Directed Pathos Total Digraphs of Some Special Arborescence Graphs

Abstract

An arborescence graph is a directed graph in which, for a vertex u called the root, and any other vertex v, there is exactly one directed path from u to v. The directed pathos of an arborescence Ar is defined as a collection of minimum number of arc disjoint open directed paths whose union is Ar. In [6], for an arborescence Ar, a directed pathos total digraph Q = DP T(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; P a 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, and it is also possible to reach the head of Pi from the tail of Pj .
In this paper, the concept of planarity of the directed pathos total digraph (that is, as an acyclic directed graph which can be drawn with non crossing arcs oriented in one direction) is being discussed and applied to a directed pathos total digraph of an arborescence Ar (DP T(Ar)). Further, the internal vertices of these directed pathos total digraph of Ar are cconsidered.
Finally, the planarity of an arborescence resulting from the vertex-gluing of two directed paths is
presented and corresponding internal vertex number is obtained.