Please solve the graph question in detail and clear.

## Question 1: Shortest Cycle Involving a Given Node.

You are given a directed graph $G: (V, E)$ using an adjacency list representation and a vertex (node) $u$ of the graph. Write an algorithm to perform the following tasks:

__1(A)__ Write an algorithm that decides (true/false) whether the vertex $u$ belongs to a cycle.

What is the complexity for your algorithm in terms of the number of vertices $|V|$ and the number of edges $|E|$?

Note: Throughout this assignment you may describe your algorithms using words and definitely use algorithms that you have already learned in class. A brief description will do.

__1(B)__ Write an algorithm which prints the smallest length cycle involving the vertex $u$.

What is the complexity for your algorithm in terms of the number of vertices $|V|$ and the number of edges $|E|$?