Graph G was created by adding two edges to tree T (29 Vertices). Inducted sub-graph H contains exactly two different cycles. How many cycles will graph G have ?
I think the catch here is "Inducted sub-graph". If I am right the result should be 2 cycles but I am not sure. Because we can add two edges to a tree and create three different cycles but we will not be able to create inducted sub-graph containing exactly 2 cycles from that graph. Am i right ?
Yes you are correct.
When 2 edges are added to a tree, either 2 disjoint cycles are created, 3 mutually overlapping cycles are created.
In the later case, all edges of each cycle present in exactly one of the remaining cycles. Thus each edge belongs to exactly two cycles. Removing any of them deletes two cycles simultaneously. Hence any induced subgraph has 0,1 or 3 cycles.
Hence it must be the former case, i.e. 2 cycles.