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On Strong Distance in Strong Oriented Graphs}

For two vertices $u$ and $v$ in
a nontrivial strong oriented graph $D$,
the strong distance $\sd(u, v)$
between $u$ and $v$ is the minimum size
of a strong subdigraph of $D$ containing $u$ and $v$.
For a vertex $v$ of $D$, the strong eccentricity $\se(v)$ is
the strong distance between $v$ and a vertex farthest from $v$.
The minimum strong eccentricity among the vertices of
$D$ is the strong radius $\srad D$,
and the maximum strong eccentricity is its strong diameter
$ \sdiam D$.
The strong radius and strong diameter of
a strong tournament are investigated. The
strong center (strong periphery) of a strong
oriented graph $D$ is
the subdigraph of $D$ induced by those vertices of strong
eccentricity $\srad(D)$ ($\sdiam (D)$). It is shown that
every oriented graph is the strong center of some strong
oriented graph. A strong oriented graph $D$ is called
strongly self-centered if
$D$ is its own strong center.
For every integer $r \geq 3$, there exist
infinitely many strongly self-centered oriented graphs
of strong radius $r$. The problem of determining
those oriented graphs that are strong peripheries of strong
oriented graphs is studied.