The vertex steiner number of a graph

Abstract

‎Let $x$ be a vertex of a connected graph $G$ and $W subset V(G)$ such that $xnotin W$‎. ‎Then $W$ is called an $x$-Steiner set of textit{G} if $W cup lbrace x rbrace$ is a Steiner set of textit{G}‎. ‎The minimum cardinality of an $x$-textit{Steiner set} of textit{G} is defined as $x$-textit{Steiner number} of textit{G} and denoted by $s_x(G)$‎. ‎Some general properties satisfied by these concepts are studied‎. ‎The $x$-textit{Steiner numbers} of certain classes of graphs are determined‎. ‎Connected graphs of order textit{p} with $x$-Steiner number 1 or $p-1$ are characterized‎. ‎It is shown that for every pair textit{a}‎, ‎textit{b} of integers with $2 leq a leq b$‎, ‎there exists a connected graph textit{G} such that $s(G)} = a$ and $s_{x}(G)=b$ for some vertex $x$ in textit{G}‎, ‎where $s(G)$ is the textit{Steiner number} of a graph‎.