In breadth-first and depth-first search, an undiscovered node is marked discovered when it is first encountered, and marked processed when it has been completely searched. At any given moment, several nodes might be simultaneously in the discovered state. (a) Describe a graph on n vertices and a particular starting vertex v such that Θ(n) nodes are simultaneously in the discovered state during a breadth-first search starting from v. (b) Describe a graph on n vertices and a particular starting vertex v such that Θ(n) nodes are simultaneously in the discovered state during a depth-first search starting from

What is the minimum number of 1’s in the adjacency matrix of a connected undirected graph of N vertices (without self–loops) ? How many 1’s are in the adjacency matrix of a complete graph of N vertices?

The graph that has been traversed can be considered acyclic if and only if the Depth-first Search (DFS) algorithm does not identify any back edges?

Run a programme to calculate the average length of the paths identified and the empirical likelihood that BreadthFirstPaths will identify a path between two randomly picked vertices for various graph models.

Please solve with the computer Question 2:  Draw a simple undirected graph G that has 11 vertices, 7 edges.

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.

3. Kleinberg, Jon. Algorithm Design (p. 519, q. 28) Consider this version of the Independent Set Problem. You are given an undirected graph G and an integer k. We will call a set of nodes I "strongly independent" if, for any two nodes v, u € I, the edge (v, u) is not present in G, and neither is there a path of two edges from u to v. That is, there is no node w such that both (v, w) and (u, w) are present. The Strongly Independent Set problem is to decide whether G has a strongly independent set of size at least k.   Show that the Strongly Independent Set Problem is NP-Complete.

For the following undirected graph (unless otherwise indicated, always visit adjacent nodes in alphabetical order.),  a. Give the adjacency matrix of the graph.  b.Give the depth-first traversal of the graph.  c.Give the breadth-first traversal of the graph.

please send correct answer for part a

Given N cities represented as vertices V₁, V2, un on an undirected graph (i.e., each edge can be traversed in both directions). The graph is fully-connected where the edge eij connecting any two vertices vį and vj is the straight-line distance between these two cities. We want to search for the shortest path from v₁ (the source) to VN (the destination). ... Assume that all edges have different values, and €₁,7 has the largest value among the edges. That is, the source and destination have the largest straight-line distance. Compare the lists of explored vertices when we run the uniform-cost search and the A* search for this problem. Hint: The straight-line distance is the shortest path between any two cities. If you do not know how to start, try to run the algorithms by hand on some small cases first; but remember to make sure your graphs satisfy the conditions in the question.