Nonseparating trees in 2-connected graphs and oriented trees in strongly connected digraphs.

Abstract Mader (2010) conjectured that for every positive integer k and every finite tree T with order m , every k -connected, finite graph G with δ (G) ≥ ⌊ 3 2 k ⌋ + m − 1 contains a subtree T ′ isomorphic to T such that G − V (T ′) is k -connected. The conjecture has been verified for paths, trees when k = 1 , and stars or double-stars when k = 2. In this paper we verify the conjecture for two classes of trees when k = 2. For digraphs, Mader (2012) conjectured that every k -connected digraph D with minimum semi-degree δ (D) = min { δ + (D) , δ − (D) } ≥ 2 k + m − 1 for a positive integer m has a dipath P of order m with κ (D − V (P)) ≥ k. The conjecture has only been verified for the dipath with m = 1 , and the dipath with m = 2 and k = 1. In this paper, we prove that every strongly connected digraph with minimum semi-degree δ (D) = min { δ + (D) , δ − (D) } ≥ m + 1 contains an oriented tree T isomorphic to some given oriented stars or double-stars with order m such that D − V (T) is still strongly connected. [ABSTRACT FROM AUTHOR]