Subject- Algortihms , Techinques and Theory

please provide an appropirate algortihm and there is no hurry so please help

1. ( 15 points) Let us define an undirected graph to be a near-tree if it is connected and has a unique (simple) cycle. Since any tree has one more vertex than it has edges, we deduce that any near-tree has an equal number of vertices and edges. Fix a graph G=(V,E). We define a subset E of E to be independent if each connected component of (V,E) is either a tree or a near-tree. Let I denote the set of all independent subsets of E. In the parts below, let X and Y be two independent sets such that X