  Fujita and Kawarabayashi [J. Combin. Theory, Ser. B 98 (2008), 805--811]
conjectured that for all positive integers $k$, $m$, there is a (least)
non-negative integer $f_{k}(m)$ such that every $k$-connected graph $G$ with
$\delta(G)\geq \lfloor\frac{3k}{2}\rfloor+ f_{k}(m)-1$ contains a connected
subgraph $W$ of order $m$ such that $G-V(W)$ is still $k$-connected. Mader
confirmed Fujita-Kawarabayashi's conjecture by proving $f_{k}(m)=m$ and $W$ is
a path. In this paper, the authors will confirm Fujita-Kawarabayashi's
conjecture again by proving $f_{k}(m)=m$ and $W$ is a spider by a new method,
where a spider is a tree with at most one vertex of degree at least three.
Meanwhile, this result will verify a conjecture proposed by Mader [J. Graph
Theory 65 (2010), 61--69] for the case of the spider.
