Consider the depth$-$first search algorithm.

visited $=$ set$()$

def dfs$($graph$,$ node$)$:

visited.add$($node$)$

for neighbor in graph$[$node$]$:

if neighbor not in visited: # assume O$(1)$ to test for membership

dfs$($graph$,$ neighbor$)$

Here, graph is a dictionary mapping nodes to sets of neighbors. For example,

graph $=[\text{'}$A$\text{'}$: $\{\text{'}$B$\text{'},\text{'}$C$\text{'}\},$

$\text{'}$B$\text{'}$: $\{\text{'}$A$\text{'},\text{'}$D$\text{'},\text{'}$E$\text{'}\},$

$...$

$]$

$($a$)$ What is the best$-$case asymptotic running time for the above function?

$($b$)$ Analogously, what is the worst$-$case asymptotic running time?

$($c$)$ Give the best possible upper bound on the asymptotic running time for arbitrary

$n.$ This should be denoted with $O,,$ or $($as appropriate$).$ Explain your answer.