Directed Pathos Total Digraph of an Arborescence
EASL-Vol. 1 (2018), Issue 1, pp. 29–42 | Open Access Full-Text PDF
M. C. Mahesh Kumar, H. M. Nagesh
Abstract:For an arborescence \(A_r\), a directed pathos total digraph \(Q=DPT(A_r)\) has vertex set \(V(Q)=V(A_r)\cup A(A_r)\cup P(A_r)\), = 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: \(ab\) such that \(a,b \in A(A_r)\) and the head of \(a\) coincides with the tail of \(b\); \(uv\) such that \(u,v \in V(A_r)\) and \(u\) is adjacent to \(v\); \(au\) \((ua)\) such that \(a\in A(A_r)\) and \(u \in V(A_r)\) and the head (tail) of \(a\) is \(u\); \(Pa\) such that \(a \in A(A_r)\) and \(P \in P(A_r)\) and the arc \(a\) lies on the directed path \(P\); \(P_iP_j\) such that \(P_i, P_j \in P(A_r)\) 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.