The minimal size of graphs with given pendant-tree connectivity

Abstract

The concept of pendant-tree $k$-connectivity $\ddot{I}{„}_k(G)$ of a graph $G$, introduced by Hager in 1985, is a generalization of classical vertex-connectivity. Let $f(n,k,\ell )$ be the minimal number of edges of a graph $G$ of order $n$ with $\ddot{I}{„}_k(G)=\ell (1\le \ell \le n-k)$. In this paper, we give some exact value or sharp bounds of the parameter $f(n,k,\ell )$.

19 pages. arXiv admin note: substantial text overlap with arXiv:1508.07202, arXiv:1508.07149, arXiv:1603.03995, arXiv:1604.01887; text overlap with arXiv:1103.6095 by other authors