Nonseparating Cycles Avoiding Specific Vertices

Thomassen proved that every k+3-connected graph G contains an induced cycle C such that G-VC is k-connected, establishing a conjecture of Lovasz. In general, one could ask the following question: For any positive integers k,l, does there exist a smallest positive integer gk,l such that for any gk,l-connected graph G, any Xi¾?VG with |X|=k, and any e∈EG-X, there is an induced cycle C in G-X such that e∈EC and G-VC is l-connected? The case when k=0 is a well-known conjecture of Lovasz that is still open for li¾?3. In this article, we prove gk,1i¾?10k+1 and gk,2i¾?10k+11. We also consider a weaker version: For any positive integers k,l, is there a smallest positive integer fk,l such that for every fk,l-connected graph G and any Xi¾?VG with |X|=k, there is an induced cycle C in G-X such that G-VC is l-connected? The case when k=0 was studied by Thomassen. We prove fk,li¾?2k+l+2 and fk,1=k+3.