Given an undirected graph G, and a path P, we want to verify that P is a cycle that contains all nodes in the graph (each node occurs on P exactly once, no repetition is allowed). In numbered steps, describe a polynomial-time verifier that checks if a given path P has the described properties (meaning that it is a cycle and it contains all nodes in the graph, with each node appearing on the path exactly once). (Hint: you can think of it in terms of this question: what does the verifier need to check to ensure that the path P has the described properties?).

One can manually count path lengths in a graph using adjacency matrices. Using the simple example below, produces the following adjacency matrix: A B A 1 1 B 1 0 This matrix means that given two vertices A and B in the graph above, there is a connection from A back to itself, and a two-way connection from A to B. To count the number of paths of length one, or direct connections in the graph, all one must do is count the number of 1s in the graph, three in this case, represented in letter notation as AA, AB, and BA. AA means that the connection starts and ends at A, AB means it starts at A and ends at B, and so on. However, counting the number of two-hop paths is a little more involved. The possibilities are AAA, ABA, and BAB, AAB, and BAA, making a total of five 2-hop paths. The 3-hop paths starting from A would be AAAA, AAAB, AABA, ABAA, and ABAB. Starting from B, the 3-hop paths are BAAA, BAAB, and BABA. Altogether, that would be eight 3-hop paths within this graph. Write a program…

Consider an undirected graph G with 100 nodes. The maximum number of edges to be included in G so that the graph is not connected is

Write a program RandomSparseGraph to generate random sparse graphs for a well-chosen set of values of V and E such that you can use it torun meaningful empirical tests on graphs drawn from the Erdös-Renyi model

Write a function in a directed graph represented by adjacency lists that returns true (1) if an edge exists between two provided vertices u and v and false (0) otherwise.

A digraph, G, is strongly connected, if each cycle in G has even length. True or False?

1. Run DFS-with-timing on this graph G: give the pre and post number of each vertex. Whenever there is a choice of vertices to explore, always pick the one that is alphabetically first. 2. Draw the meta-graph of G. 3. What is the minimum number of edges you must add to G to make it strongly connected (i.e., it consists of a single connected component after adding these edges)? Give such a set of edges. b.

5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…

Can you help me solve this exercise? Please note that the greedy approach described in the advice paragraph does not work.

I have a directed graph with N nodes.  How can I tell if the graph has a cycle?   Your answer should be an algorithm in English.  It is nice but not essential for you to give the name of the algorithm. You should describe how the algorithm works.