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)≥⌊32k⌋+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)}≥2k+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.