University of IsfahanTransactions on Combinatorics2251-86579220200601The vertex Steiner number of a graph1151242458010.22108/toc.2020.116191.1628ENJ.JOHNDepartment 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 $x\notin W$. Then $W$ is called an $x$-Steiner set of <em>G</em> if $W \cup \lbrace x \rbrace$ is a Steiner set of <em>G</em>. The minimum cardinality of an $x$-<em>Steiner set</em> of <em>G</em> is defined as $x$-<em>Steiner number</em> of <em>G</em> and denoted by $s_x(G)$. Some general properties satisfied by these concepts are studied. The $x$-<em>Steiner numbers</em> of certain classes of graphs are determined. Connected graphs of order <em>p</em> with $x$-Steiner number 1 or $p-1$ are characterized. It is shown that for every pair <em>a</em>, <em>b</em> of integers with $2 \leq a \leq b$, there exists a connected graph <em>G</em> such that $s(G)} = a$ and $s_{x}(G)=b$ for some vertex $x$ in <em>G</em>, where $s(G)$ is the <em>Steiner number</em> of a graph.https://toc.ui.ac.ir/article_24580_0d5028a2912e9c2fbf79c364b27d26e3.pdf