A note on the restricted arc connectivity of oriented graphs of girth four

Let $D$ be a strongly connected digraph. An arc set $S$ of $D$ is a \emph{restricted arc-cut} of $D$ if $D-S$ has a non-trivial strong component $D_1$ such that $D-V(D_1)$ contains an arc. The \emph{restricted arc-connectivity} ${\lambda }^{\mathrm{\prime }}(D)$ of a digraph $D$ is the minimum cardinality over all restricted arc-cuts of $D$. A strongly connected digraph $D$ is \emph{${\lambda }^{\mathrm{\prime }}$-connected} when ${\lambda }^{\mathrm{\prime }}(D)$ exists. This paper presents a family $\mathcal{F}$ of strong digraphs of girth four that are not ${\lambda }^{\mathrm{\prime }}$-connected and for every strong digraph $D\mathrm{\notin }\mathcal{F}$ with girth four it follows that it is ${\lambda }^{\mathrm{\prime }}$-connected. Also, an upper and lower bound for ${\lambda }^{\mathrm{\prime }}(D)$ are given.