The Transformation Digraph $D^{++-}$

Abstract

Let $D$ be a digraph with order at least two. The transformation digraph $D^{++-}$ is the digraph with vertex set $V(D)\cup A(D)$ in which $(x,y)$ is an arc of $D^{++-}$ if one of the following conditions holds:(i) $x,y\in V(D)$, and $(x,y)$ is an arc of $D$;(ii) $x,y\in A(D)$, and the head of $x$ is the tail of $y$;(iii) $x\in V(D),y\in A(D)$, and $x$ is not the tail of $y$;(iv) $x\in A(D),y\in V(D)$, and $y$ is not the head of $x$.In this paper, we determine the regularity and diameter of $D^{++-}$. Furthermore, we characterize maximally-arc-connected or super-arc-connected $D^{++-}$. We also give sufficient conditions for this kind of transformation digraph to be maximally-connected or super-connected.