Publications results for "Contents of: Acta Universitatis Sapientiae. Mathematica. An International Journal of the Sapientia University [2 (2010), no. 2]"

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**MR2748464**

**(2012a:05097)**Reviewed

Santhakumaran, A. P.(6-STXAP-PGM); Titus, P.

The connected vertex detour number of a graph. (English summary)

*Acta Univ. Sapientiae Math.*2 (2010), no. 2, 146–159.

05C12

Publication Year 2010
Review Published2011-10-13

Summary: "For a connected graph $G$ of order $p\geq2$ and a vertex $x$
of $G$, a set $S\subseteq V(G)$ is an $x$-detour set of $G$ if each
vertex $v\in V(G)$ lies on an $x-y$ detour for some element $y$ in $S$.
The minimum cardinality of an $x$-detour set of $G$ is defined as the
$x$-detour number of $G$, denoted by $d_x(G)$. An $x$-detour set of
cardinality $d_x(G)$ is called a $d_x$-set of $G$. A connected
$x$-detour set of $G$ is an $x$-detour set $S$ such that the subgraph
$G[S]$ induced by $S$ is connected. The minimum cardinality of a
connected $x$-detour set of $G$ is defined as the connected $x$-detour
number of $G$ and is denoted by $cd_x(G)$. A connected $x$-detour set
of cardinality $cd_x(G)$ is called a $cd_x$-set of $G$. We determine
bounds for the connected $x$-detour number and find the same for some
special classes of graphs. If $a,\ b$ and $c$ are positive integers
such that $3\leq a\leq b+1<c$, then there exists a connected graph $G$
with detour number $dn(G)=a,\ d_x(G)=b$ and $cd_x(G)=c$ for some vertex
$x$ in $G$. For positive integers $R,\ D$ and $n\geq3$ with
$R<D\leq2R$, there exists a connected graph $G$ with ${\rm rad}_DG=R$,
${\rm diam}_DG=D$ and $cd_x(G)=n$ for some vertex $x$ in $G$. Also, for each
triple $D,\ n$ and $p$ of integers with $4\leq D\leq p-1$ and $3\leq
n\leq p$, there is a connected graph $G$ of order $p$, detour diameter
$D$ and $cd_x(G)=n$ for some vertex $x$ of $G$.''