Abstract

For a connected graph $G$, a decomposition $\pi=\{G_1, G_2,..., G_n\}$ is called a total Steiner decomposition if $s_t(G_i )=s_t(G)$ for $1\leq i\leq n$. The maximum cardinality obtained for the total Steiner decomposition $\pi$ is said to be total Steiner decomposition number of $G$ and is denoted by $\pi_{tst}(G)$. In this paper, we present some properties of $\pi_{tst}(G)$. Also it is shown that for every pair of positive integers $m, n$ with $m\geq4$, there exists a connected graph $G$ such that $s_t(G)=m$ and $\pi_{tst}(G)=n$.

Keywords and Phrases

Total Steiner number, total Steiner decomposition number, Realization theorem.

A.M.S. subject classification

05C12, 05C99.

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