Note on the connectivity keeping spiders in $k$-connected graphs

Abstract

W. Mader [J. Graph Theory 65 (2010), 61--69] conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with $\hat{I}´(G)\ge \lfloor \frac{3k}{2}\rfloor +m-1$ contains a tree $T^{\prime }\cong T$ such that $G-V(T^{\prime })$ remains $k$-connected. In 2010, Mader confirmed the conjecture for the $k$-connected graph if $T$ is a path; very recently, Liu et al. confirmed the conjecture if $k=2,3$. The conjecture is open for $k\ge 4$ till now. In this paper, we show that Mader's conjecture is true for the $k+1$-connected graph if $T$ is a spider and $\hat{I}”(G)=|G|-1$.

11 pages