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

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.