Article

 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: [bf] T. Bonnesen and W. Fenchel: Theorie der konvexen Körper. Springer, Berlin, 1934; transl. by L. Boron, C. Christenson, and B. Smith: Theory of Convex Bodies. BCS Associates, Moscow, ID, 1987.. MR 0920366 Reference: [bh:dg] F. Buckley and F. Harary: Distance in Graphs.Addison-Wesley, Redwood City, CA, 1990. MR 1045632 Reference: [BH2] F. Buckley and F. Harary: Closed geodetic games for graphs.Congr. Numer. 47 (1985), 131–138. MR 0830675 Reference: [BH3] F. Buckley and F. 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