The vertex steiner number of a graph
J. John · DOAJ (DOAJ: Directory of Open Access Journals) · 2020
Let $x$ be a vertex of a connected graph $G$ and $W \subset V(G)$ such that $x otin W$. 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.