# The vertex steiner number of a graph

Document Type : Research Paper

Author

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

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 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.

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