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

TL;DR

This paper investigates the restricted arc connectivity of strongly connected digraphs with girth four, identifying a family of exceptions and establishing bounds for the connectivity measure.

Contribution

It introduces a family of girth four digraphs that are not restricted arc-connected and proves that all others are, providing bounds for the restricted arc connectivity.

Findings

Identifies a specific family of girth four digraphs that are not λ'-connected.

Proves that all other strong digraphs of girth four are λ'-connected.

Provides upper and lower bounds for the restricted arc connectivity λ'.

Abstract

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} $\lambda'(D)$ of a digraph $D$ is the minimum cardinality over all restricted arc-cuts of $D$. A strongly connected digraph $D$ is \emph{$\lambda'$-connected} when $\lambda'(D)$ exists. This paper presents a family $\cal{F}$ of strong digraphs of girth four that are not $\lambda'$-connected and for every strong digraph $D\notin \cal{F}$ with girth four it follows that it is $\lambda'$-connected. Also, an upper and lower bound for $\lambda'(D)$ are given.