$10\%$

$($a$)$ Consider the following graph and answer the following questions.

$($i$)$ List two possible sequences of vertices being visited in the order of

a breath$-$first search starting from $v.$

$($ii$)$ List two possible sequences of vertices being visited in the order of

a depth$-$first search starting from $v4.$

$($b$)$ Assuming that we use the adjacent list to represent a simple undirected

graph, design an algorithm to report if the given graph contains a cycle or

not and analyze the time complexity of your algorithm.