graph algorithms - our solution to reporting path from to in code fragment could be made more efficient in practice if the dfs process ended as soon as is discovered describe how to modify our code base to implement this optimization - let be an undirected graph with vertices and edges describe an ( )-time algorithm for traversing each edge of exactly once in each direction if one - implement an algorithm that returns cycle in directed graph gexists - write functioncomponents( )for undirected graph gthat returns dictionary mapping each vertex to an integer that serves as an identifier for its connected component that istwo vertices should be mapped to the same identifier if and only if they are in the same connected component - say that maze is constructed correctly if there is one path from the start to the finishthe entire maze is reachable from the startand there are no loops around any portions of the maze given maze drawn in an gridhow can we determine if it is constructed correctlywhat is the running time of this algorithmc- computer networks should avoid single points of failurethat isnetwork vertices that can disconnect the network if they fail we say an undirectedconnected graph is biconnected if it contains no vertex whose removal would divide into two or more connected components give an algorithm for adding at most edges to connected graph gwith > vertices and > edgesto guarantee that is biconnected your algorithm should run in ( mtime - explain why all nontree edges are cross edgeswith respect to bfs tree constructed for an undirected graph - explain why there are no forward nontree edges with respect to bfs tree constructed for directed graph - show that if is bfs tree produced for connected graph gthenfor each vertex at level ithe path of between and has edgesand any other path of between and has at least edges - justify proposition - provide an implementation of the bfs algorithm that uses fifo queuerather than level-by-level formulationto manage vertices that have been discovered until the time when their neighbors are considered - graph is bipartite if its vertices can be partitioned into two sets and such that every edge in has one end vertex in and the other in design and analyze an efficient algorithm for determining if an undirected graph is bipartite (without knowing the sets and in advance