On the pedant tree-connectivity of graphs

Abstract

The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then this tree is called a \emph{pedant $S$-Steiner tree}. Two pedant $S$-Steiner trees $T$ and $T'$ are said to be \emph{internally disjoint} if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. For $S\subseteq V(G)$ and $|S|\geq 2$, the \emph{local pedant-tree connectivity} $\tau_G(S)$ is the maximum number of internally disjoint pedant $S$-Steiner trees in $G$. For an integer $k$ with $2\leq k\leq n$, \emph{$k$-pedant tree-connectivity} is defined as $\tau_k(G)=\min\{\tau_G(S)\,|\,S\subseteq V(G),|S|=k\}$. In this paper, we first study the sharp bounds of pedant tree-connectivity. Next, we obtain the exact value of a threshold graph, and give an upper bound of the pedant-tree $k$-connectivity of a complete multipartite graph. For a connected graph $G$, we show that $0\leq \tau_k(G)\leq n-k$, and graphs with $\tau_k(G)=n-k,n-k-1,n-k-2,0$ are characterized in this paper. In the end, we obtain the Nordhaus-Guddum type results for pedant tree-connectivity.