Recall the BFS algorithm that runs on graph $\backslash ($ G $\backslash )$ starting from root node $\backslash ($ s $\backslash$in V $\backslash )$ :

$1.$ Consider the undirected graph shown in Figure $1.$ If we begin a BFS traversal starting at node $\backslash ($ A $\backslash ),$ in what order are the nodes visited? Assume that the adjacency list is ordered alphabetically, i$.$e$.,$ we visit earlier letters alphabetically first, so from $\backslash ($ A $\backslash )$ we will visit $\backslash ($ B $\backslash )$ before $\backslash ($ C $\backslash ),$ and so on$.$

$2.$ Explain what happens if we don't check whether a node is white before enqueuing it$.$

$3.$ Assume we replace the $($FIFO$)$ queue with a stack and therefore in line $7$ obtain the node last added. Does $\backslash ($ v $\backslash ).$dist still represent the shortest distance from $\backslash ($ s $\backslash )$ to $\backslash ($ v $\backslash )?$

$4.$ Consider graphs for which edges have assigned weights. Assume we modify line $11$ of BFS to add the weight of the edge from $\backslash ($ u $\backslash )$ to $\backslash ($ v $\backslash )$ instead of adding $1.$ Does the modified algorithm compute the shortest weighted distance correctly? If so$,$ prove the correctness; otherwise, provide a counterexample for which the algorithm fails.