# The vertex steiner number of a graph

Document Type : Research Paper

Author

Department of Mathematics, Government college of Engineering, Tirunelveli, India- 627007

10.22108/toc.2020.116191.1628

Abstract

‎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 \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.

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