| Title:
|
Extreme geodesic graphs (English) |
| Author:
|
Chartrand, Gary |
| Author:
|
Zhang, Ping |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2002 |
| Pages:
|
771-780 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For two vertices $u$ and $v$ of a graph $G$, the closed interval $I[u, v]$ consists of $u$, $v$, and all vertices lying in some $u\text{--}v$ geodesic of $G$, while for $S \subseteq V(G)$, the set $I[S]$ is the union of all sets $I[u, v]$ for $u, v \in S$. A set $S$ of vertices of $G$ for which $I[S]=V(G)$ is a geodetic set for $G$, and the minimum cardinality of a geodetic set is the geodetic number $g(G)$. A vertex $v$ in $G$ is an extreme vertex if the subgraph induced by its neighborhood is complete. The number of extreme vertices in $G$ is its extreme order $\mathop {\mathrm ex}(G)$. A graph $G$ is an extreme geodesic graph if $g(G) = \mathop {\mathrm ex}(G)$, that is, if every vertex lies on a $u\text{--}v$ geodesic for some pair $u$, $ v$ of extreme vertices. It is shown that every pair $a$, $ b$ of integers with $0 \le a \le b$ is realizable as the extreme order and geodetic number, respectively, of some graph. For positive integers $r, d,$ and $k \ge 2$, it is shown that there exists an extreme geodesic graph $G$ of radius $r$, diameter $d$, and geodetic number $k$. Also, for integers $n$, $ d, $ and $k$ with $2 \le d < n$, $2 \le k < n$, and $n -d - k +1 \ge 0$, there exists a connected extreme geodesic graph $G$ of order $n$, diameter $d$, and geodetic number $k$. We show that every graph of order $n$ with geodetic number $n-1$ is an extreme geodesic graph. On the other hand, for every pair $k$, $ n$ of integers with $2 \le k \le n-2$, there exists a connected graph of order $n$ with geodetic number $k$ that is not an extreme geodesic graph. (English) |
| Keyword:
|
geodetic set |
| Keyword:
|
geodetic number |
| Keyword:
|
extreme order |
| Keyword:
|
extreme geodesic graph |
| MSC:
|
05C12 |
| MSC:
|
05C35 |
| idZBL:
|
Zbl 1009.05051 |
| idMR:
|
MR1940058 |
| . |
| Date available:
|
2009-09-24T10:56:37Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127763 |
| . |
| Reference:
|
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