*(English)*Zbl 1235.05045

Summary: For a connected graph $G$ of order $p\ge 2$ and a vertex $x$ of $G$, a set $S\subseteq V\left(G\right)$ is an $x$-detour set of $G$ if each vertex $v\in V\left(G\right)$ 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 ${d}_{x}\left(G\right)$. An $x$-detour set of cardinality ${d}_{x}\left(G\right)$ 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 $G\left[S\right]$ 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 $cd+x\left(G\right)$. A connected $x$-detour set of cardinality $c{d}_{x}\left(G\right)$ is called a $c{d}_{x}$-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 $3\le a\le b+1<c$, then there exists a connected graph $G$ with detour number $dn\left(G\right)=a$, ${d}_{x}\left(G\right)=b$ and $c{d}_{x}\left(G\right)=c$ for some vertex $x$ in $G$.

For positive integers $R$, $D$ and $n\ge 3$ with $R<D<2R$, there exists a connected graph $G$ with ${\text{rad}}_{D}G=R$, ${\text{diam}}_{D}G=D$ and $c{d}_{x}\left(G\right)=n$ for some vertex $x$ in $G$. Also, for each triple $D$, $n$ and $p$ of integers with $4\le D\le p-1$ and $3\le n\le p$, there is a connected graph $G$ of order $p$, detour diameter $D$ and $c{d}_{x}\left(G\right)=n$ for some vertex $x$ of $G$.