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.