Digraphs with Isomorphic Underlying and Domination Graphs: Pairs of Paths

Abstract

The domination graph of a digraph $D$, denoted $\text{dom}(D)$, is created using the vertex set of $D$ and edge $uv\in E(\text{dom}(D))$ whenever $(u,z)\in A(D)$ or $(v,z)\in A(D)$ for any other vertex $z\in V(D)$. Specifically, we consider directed graphs whose underlying graphs are isomorphic to their domination graphs. In particular, digraphs are completely characterized where $U{G}^c(D)$ is the union of two disjoint paths.