(t,k)-Shredders in k-Connected Graphs

Abstract

Let $t$, $k$ be integers with $t\ge 3$ and $k\ge 1$. For a graph $G$, a subset $S$ of $V(G)$ with cardinality $k$ is called a $(t,k)$-shredder if $G-S$ consists of $t$ or more components. In this paper, we show that if $t\ge 3$, $2(t-1)\le k\le 3t-5$ and $G$ is a $k$-connected graph of order at least $k^8$, then the number of $(t,k)$-shredders of $G$ is less than or equal to $((2t-1)(|V(G)|-f(|V(G)|)))/(2(t-1)2)$, where $f(n)$ denotes the unique real number $x$ with $x\ge k-1$ such that $n=2{(t-1)}^2{(}_k^x)+x$.