%0 Journal Article
%T The vertex Steiner number of a graph
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A JOHN, J.
%D 2020
%\ 06/01/2020
%V 9
%N 2
%P 115-124
%! The vertex Steiner number of a graph
%K Steiner distance
%K Steiner number
%K vertex Steiner number
%R 10.22108/toc.2020.116191.1628
%X Let $x$ 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 G if $W \cup \lbrace x \rbrace$ is a Steiner set of G. The minimum cardinality of an $x$-Steiner set of G is defined as $x$-Steiner number of G and denoted by $s_x(G)$. Some general properties satisfied by these concepts are studied. The $x$-Steiner numbers of certain classes of graphs are determined. Connected graphs of order p with $x$-Steiner number 1 or $p-1$ are characterized. It is shown that for every pair a, b of integers with $2 \leq a \leq b$, there exists a connected graph G such that $s(G)} = a$ and $s_{x}(G)=b$ for some vertex $x$ in G, where $s(G)$ is the Steiner number of a graph.
%U https://toc.ui.ac.ir/article_24580_0d5028a2912e9c2fbf79c364b27d26e3.pdf