On the Connectivity Keeping some Trees in Connected and 2-Connected Bipartite Graphs

In 2010, Mader conjectured that for any tree T of order m, every k-connected graph G with minimum degree at least $\lfloor \frac{3k}{2}\rfloor$+m-1 contains a subtree $T^{\prime }\cong T$ such that G−V(T′) is still k-connected. For bipartite graphs, we proposed a similar conjecture as follows: 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 minimum degree at least k+t contains a subtree $T^{\prime }\cong T$ such that G−V (T′) is still k-connected. In this paper, we confirm this conjecture for all caterpillars whose internal vertices is at most 3 when k=1 and k=2.