Assignment 4 Theoretical Part Solution

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Description

Note: your solution must be submitted as a single pdf file

Question 1. [Graphs]

Let be the undirected graph with vertices = {0,1,2,3,4,5,6,7,8} and edges = {{0,1}, {0,4}, {0,5}, {1,2}, {1,5}, {1,6}, {2,3}, {2,6}, {3,6}, {3,7}, {4,8}, {5,6}, {5,8}, {6,8}, {7,8}}

  1. Draw in such a way that no two edges cross (A graph that can be depicted without edge crossing is called a planar graph.)

  1. Show an adjacency list representation of .

  1. What is the adjacency matrix representation of ? Show the adjacency matrix where a 1 represents the existence of an edge and 0 denotes non-adjacent vertex pairs.

  1. For the graph in Question 1, assume that in a traversal of , the adjacent vertices of a given vertex are returned in their numeric order.

    1. Order the vertices as they are visited in a DFS traversal starting at vertex 0.

    1. Order the vertices as they are visited in a BFS traversal starting at vertex 0.

Question 2. [Graph Properties]

Let undirected simple graph = ( , ) be a forest with vertices, edges and connected components. Prove that the number of edges in is = − .

Question 3. [Topological Sort]

  1. In pseudocode, design an algorithm that determines whether a digraph has a unique topological ordering. Your algorithm should return the ordering if a unique one exists, and indicate that no unique topological order, or no topological order exists, otherwise.

  1. Are the following statements true or false? Argue convincingly.

    1. A postorder traversal on a tree always produces a topological ordering. For this, assume that consider the tree as directed graph: there is a directed edge (child, parent) between each child and parent [that is, the child is the source, and the parent the destination of the directed edge].

    1. If a graph has a topological ordering, then a depth-first traversal of the same directed graph will not see any back edges.

Question 4. [Describing problems as computational problems]

Consider a social network. The goal is to find in the in the network a largest group of people who are all friends with each other. Describe the problem as a graph problem. Clearly state input and output. Explain why solving your graph problem will solve the original problem.