$25.$ Let G be a weighted graph in which the weight of each edge is a positive integer. Let G$$ be the graph obtained from G by replacing each edge in G of weight k by k unweighted edges in series: Show that Dijkstra$$s algorithm for finding the minimum length of each path in the weighted graph G from a fixed vertex v to all other vertices $($Algorithm $8\mathrm{.}4\mathrm{.}1)$ and performing a breadth$-$first search in the unweighted graph G$$ starting with vertex v are, in effect, the same process. Show th Document reading pane rithm for finding the minimum length of each path in the weighted graph G from a fixed vertex v to all other vertices $($Algorithm $8\mathrm{.}4\mathrm{.}1)$ and performing a breadth$-$first search in the unweighted graph G$^(\text{'})$ starting with vertex v are, in effect, the same process.

$31.$ Write a depth$-$first search algorithm to test whether a graph is connected.

$32.$ Write a depth$-$first search algorithm that finds all solutions to the four$-$queens

problem.