Connectivity keeping caterpillars and spiders in bipartite graphs with connectivity at most three

Abstract

A conjecture of Luo, Tian and Wu (2022) says that for every positive integer $k$ and every finite tree $T$ with bipartition $X$ and $Y$ (denote $t=\max \{|X|,|Y|\})$, every $k$-connected bipartite graph $G$ with $\hat{I}´(G)\ge k+t$ contains a subtree $T^{\prime }\cong T$ such that $\hat{I}º(G-V(T^{\prime }))\ge k$. In this paper, we confirm this conjecture for caterpillars when $k=3$ and spiders when $k\le 3$.