A Cayley graph (resp. digraph) $Cay(G,S)$ is called a $CI$-graph (resp. $DCI$-graph) of $G$ if, for any Cayley graph (resp. digraph) $Cay(G, T)$, whenever $Cay(G,S) \cong Cay(G, T)$ we have $S = T^\sigma$ for some $\sigma \in Aut(G)$. A group $G$ is called a $CI$-group (resp. $DCI$-group) if all Cayley graphs (resp. digraphs) of $G$ are $CI$-graphs (resp. $DCI$-graphs). 


It is well known that $D\_{2p}$ ($p$ is a prime) is a $CI$-group (see *Babai, L.: Isomorphism problem for a class of point-symmetric structures. Acta Math. Acad. Sci. Hung. 29, 329–336 (1977)*). 


I want to ask whether $D\_{2p}$ is a also a $DCI$-group. 



DCI-properties of Cayley graphs
Corollary 4.5 of Babai's paper states that $D\_{2p}$ is CI for the category of "colour-graphs". Looking at the definition on page 330, you can see that Babai does not require "colour-graphs" to be undirected. So his result applies to digraphs, and therefore tells us that $D\_{2p}$ is a DCI-group.


