Directed pathos middle digraph of an arborescence

Abstract

A directed pathos middle digraph of an arborescence A?, written Q = DPM(A?), is the digraph whose vertex set V (Q) = V (A?) ? A(A?) ? P(A?), where V (A?) is the vertex set, A(A?) is the arc set, and P(A?) is a directed pathos set of A?. The arc set A(Q) consists of the following arcs: ab such that a, b ? A(A?) and the head of a coincides with the tail of b; for every v ? V (A?), all arcs a1v, va2; for which v is a head of the arc a1 and tail of the arc a2 in A?; P a such that a ? A(A?) and P ? P(A?) and the arc a lies on the directed path P; P?iPj such that Pi, Pj ? P(A?) 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 . The problem of reconstructing an arborescence from its directed pathos middle digraph is presented. The characterization of digraphs whose DPM(A?) are planar; outerplanar; maximal outerplanar; and minimally non-outerplanar is studied.