I. The input is a connected undirected graph G, a spanning tree T of G, and a vertex u V(G). Design an O(El) time algorithm to determine whether T is a valid DFS tree of G rooted at v, .e., determine whether T can be an output of DFS under some order of edges starting with v. 2. Characterize conditions of undirected graphs that contain a vertex v such that there exists a DFS tree rooted at that is identical to a BFS tree rooted at v. Note that two spanning trees are identical if they contain the same set of edges 3. Given a undirected graph G, the length (no weight) of the smallest cycle is called the girth of the graph. Design an O(lEl) time algorithm to compute the girth of a given graph G. (u2,U2), (U3,U3) of E(G), Given a donnected undirected graph G and three edges (ui,n), design an O(IE1) time algorithm to determine whether there exists a cycle in G that contains both (ul,tn) and (u2,tz), but does not contain (u3,V3). 4