Connectivity keeping stars or double-stars in 2-connected graphs

TL;DR

This paper proves Mader's conjecture for the case when the tree is a star or double-star and the graph is 2-connected, extending previous results that confirmed the conjecture for paths and trees when k=1.

Contribution

The paper establishes the validity of Mader's conjecture specifically for stars and double-stars in 2-connected graphs, a case not previously confirmed.

Findings

Mader's conjecture holds for stars and double-stars when k=2.

The result extends the class of trees for which the conjecture is verified.

The proof applies to finite, 2-connected graphs with specified minimum degree.

Abstract

In [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.], Mader conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $\delta(G)\geq \lfloor\frac{3}{2}k\rfloor+m-1$ contains a subtree $T'$ isomorphic to $T$ such that $G-V(T')$ is $k$-connected. In the same paper, Mader proved that the conjecture is true when $T$ is a path. Diwan and Tholiya [A.A. Diwan, N.P. Tholiya, Non-separating trees in connected graphs, Discrete Math. 309 (2009) 5235-5237.] verified the conjecture when $k=1$. In this paper, we will prove that Mader's conjecture is true when $T$ is a star or double-star and $k=2$.