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

TL;DR

This paper proves Mader's conjecture for the case where the graph is $(k+1)$-connected, the tree is a spider, and the graph has maximum degree one less than the number of vertices, extending previous results.

Contribution

It confirms Mader's conjecture for spider trees in $(k+1)$-connected graphs with maximum degree $|G|-1$, a case previously unresolved.

Findings

Mader's conjecture holds for spider trees under specified conditions.

The result applies to $(k+1)$-connected graphs with maximum degree $|G|-1$.

Extends known cases where the conjecture is confirmed.

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 $\delta(G)\geq\lfloor\frac{3k}{2}\rfloor+m-1$ contains a tree $T'\cong T$ such that $G-V(T')$ 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 $k\geq 4$ 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 $\Delta(G)=|G|-1$.