A smallest irregular oriented graph containing a given diregular one (Q1883255)

Let \(D\) be an oriented graph. Then \(D\) is regular if there is a number \(r\) so that \(\text{od}(v)= \text{id}(v)=r\) for each vertex \(v\) of \(D\); and \(D\) is irregular if for any two vertices \(v\neq w\) of \(D\) we have \((\text{od}(v), \text{id}(v))\neq (\text{od}(w),\text{id}(w))\). In the paper, for any oriented regular graph \(D\), a smallest irregular oriented graph \(F\) is constructed such that \(F\) includes \(D\) as induced subgraph. Further, it is shown that the total number of irregular oriented graphs is superexponential in their order.