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?).

We are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…

The Generalized Voronoi Graph may be used to map space, therefore come up with an exploration method and describe how it works!

The following is a graph theory problem: Suppose we have a graph A = (X, Y) and a subgraph of A, which is expressed as B = (x, y). x is an element of X and y is an element of Y. A subgraph is plainly described as a graph obtained from A by removing vertices &/or edges. For every following statement determine whether it is true or false. Explain or provide counterexample. Be careful when answering the following questions to leave edges that have their endpoints in B.  (i) If A is connected then B is connected (and vice versa: If B... then A) (ii)  If A is simple graph then B is simple graph (and vice versa: If B... then A) (iii) If A contains a cycle then B contains a cycle (and vice versa: If B... then A) (iv) If A is a tree then B is a tree (and vice versa: If B... then A)

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…

Computer Science

Every pair of vertices in a graph that is linked by two different paths is said to be biconnected. An articulation point in a connected graph is a vertex that, if it and its surrounding edges were eliminated, would cause the graph to become disconnected. demonstrate the biconnection of any graph lacking articulation points. Tip: To create two disjoint paths connecting s and t given a set of vertices s and t and a path connecting them, take advantage of the fact that none of the vertices on the path are articulation points.

Please help with this question:

Adjacency lists on a graph may be represented as linked lists, but only in a certain form. Show your argument with an example. Is it true that no prior experience or education is necessary to learn how to code?

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