University of IsfahanTransactions on Combinatorics2251-86579220200601The vertex steiner number of a graph1151242458010.22108/toc.2020.116191.1628ENJJOHNDepartment of Mathematics, Government college of Engineering, Tirunelveli, India- 627007Journal Article20190324Let $x$ 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.https://toc.ui.ac.ir/article_24580_0d5028a2912e9c2fbf79c364b27d26e3.pdf