Strong Distance in Strong Digraphs



For two vertices $u$ and $v$ in a strong oriented graph $D$ of order $n \geq 3$, 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$. It is shown that every pair $r, d$ of integers with $3 \leq r \leq d \leq 2r$ is realizable as the strong radius and strong diameter of some strong oriented graph. Also, for every strong oriented graph $D$ of order $n \geq3$, it is shown that $\sdiam (D) \leq \lfloor{5(n-1)/3}\rfloor$. Furthermore, for every integer $n \geq 3$, there exists a strong oriented graph $D$ of order $n$ such that $\sdiam (D) = \lfloor{5(n-1)/3}\rfloor$.