For a nonempty set $S$ of vertices in a strong digraph $D$, the strong distance $d(S)$ is the minimum size of a strong subdigraph of $D$ containing the vertices of $S$. If $S$ contains $k$ vertices, then $d(S)$ is referred to as the $k$-strong distance of $S$. For an integer $k \ge 2$ and a vertex $v$ of a strong digraph $D$, the $k$-strong eccentricity $\mathop {\mathrm se}_k(v)$ of $v$ is the maximum $k$-strong distance $d(S)$ among all sets $S$ of $k$ vertices in $D$ containing $v$. The minimum $k$-strong eccentricity among the vertices of $D$ is its $k$-strong radius $\mathop {\mathrm srad}_k D$ and the maximum $k$-strong eccentricity is its $k$-strong diameter $_k D$. The $k$-strong center ($k$-strong periphery) of $D$ is the subdigraph of $D$ induced by those vertices of $k$-strong eccentricity $\mathop {\mathrm srad}_k(D)$ ($_k (D)$). It is shown that, for each integer $k \ge 2$, every oriented graph is the $k$-strong center of some strong oriented graph. A strong oriented graph $D$ is called strongly $k$-self-centered if $D$ is its own $k$-strong center. For every integer $r \ge 6$, there exist infinitely many strongly 3-self-centered oriented graphs of 3-strong radius $r$. The problem of determining those oriented graphs that are $k$-strong peripheries of strong oriented graphs is studied.
For a nonempty set $S$ of vertices in a strong digraph $D$, the strong distance $d(S)$ is the minimum size of a strong subdigraph of $D$ containing the vertices of $S$. If $S$ contains $k$ vertices, then $d(S)$ is referred to as the $k$-strong distance of $S$. For an integer $k \ge 2$ and a vertex $v$ of a strong digraph $D$, the $k$-strong eccentricity $\mathop {\mathrm se}_k(v)$ of $v$ is the maximum $k$-strong distance $d(S)$ among all sets $S$ of $k$ vertices in $D$ containing $v$. The minimum $k$-strong eccentricity among the vertices of $D$ is its $k$-strong radius $\mathop {\mathrm srad}_k D$ and the maximum $k$-strong eccentricity is its $k$-strong diameter $_k D$. The $k$-strong center ($k$-strong periphery) of $D$ is the subdigraph of $D$ induced by those vertices of $k$-strong eccentricity $\mathop {\mathrm srad}_k(D)$ ($_k (D)$). It is shown that, for each integer $k \ge 2$, every oriented graph is the $k$-strong center of some strong oriented graph. A strong oriented graph $D$ is called strongly $k$-self-centered if $D$ is its own $k$-strong center. For every integer $r \ge 6$, there exist infinitely many strongly 3-self-centered oriented graphs of 3-strong radius $r$. The problem of determining those oriented graphs that are $k$-strong peripheries of strong oriented graphs is studied.
@article{10_21136_MB_2002_133957,
author = {Zhang, Ping},
title = {On $k$-strong distance in strong digraphs},
journal = {Mathematica Bohemica},
pages = {557--570},
year = {2002},
volume = {127},
number = {4},
doi = {10.21136/MB.2002.133957},
mrnumber = {1942641},
zbl = {1003.05037},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2002.133957/}
}
TY - JOUR
AU - Zhang, Ping
TI - On $k$-strong distance in strong digraphs
JO - Mathematica Bohemica
PY - 2002
SP - 557
EP - 570
VL - 127
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.21136/MB.2002.133957/
DO - 10.21136/MB.2002.133957
LA - en
ID - 10_21136_MB_2002_133957
ER -
