4. Let nЄ Z+ with n ≥ 4, and let the vertex set V' for the complete graph K-1 be {V1, V2, V3, ..., Un−1}. Now construct the loop-free undirected graph G = (V,E) from KË−1 as follows: V = V'U{v}, and E consists of all the edges in K-1 except for the edge {v1, v2}, which is replaced by the pair of edges {v1, v} and {v, v2}. a) Determine deg(x) + deg(y) for all nonadjacent vertices x and y in V. b) Does G have a Hamilton cycle? c) How large is the edge set E? d) Do the results in parts (b) and (c) contradict Corollary 11.6? Corollary 11.6: If G = (V,E) is a loop-free undirected graph with |V| = n ≥ 3, and if |E| ≥ ("₂¹) +2, then G has a Hamilton cycle.

Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k.

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…

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Construct all regular graphs that exist for the vertex set V = {1, 2, 3, 4}. For example, G = (V,E0) where E0 = {} is one possible graph

To help jog your memory, here are some definitions: Vertex Cover: given an undirected unweighted graph G = (V, E), a vertex cover Cy of G is a subset of vertices such that for every edge e = (u, v) = E, at least one of u or v must be in the vertex cover Cy. Set Cover: given a universe of elements U and a collection of sets S = {S₁, ..., Sm}, a set cover is any (sub)collection C's whose union equals U. In the minimum vertex cover problem, we are given an undirected unweighted graph G = (V, E), and are asked to find the smallest vertex cover. For example, in the following graph, {A, E, C, D} is a vertex cover, but not a minimum vertex cover. The minimum vertex covers are {B, E, C} and {A, E, C}. A B E с F D Then, recall in the minimum set cover problem, we are given a set U and a collection S = {S₁, ..., Sm} of subsets of U, and are asked to find the smallest set cover. For example, given U := {a, b, c, d}, S₁ := {a, b,c}, S₂ = = {b,c}, and S3 := {c, d), a solution to the problem is C's…

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

How do I do this?  We say a graph G = (V, E) has a k-coloring for some positive integer k if we can assign k different colors to vertices of G such that for every edge (v, w) ∈ E, the color of v is different to the color w. More formally, G = (V, E) has a k-coloring if there is a function f : V → {1, 2, . . . , k} such that for every (v, w) ∈ E, f(v) 6= f(w).3-Color problem is defined as follows: Given a graph G = (V, E), does it have a 3-coloring?4-Color problem is defined as follows: Given a graph G = (V, E), does it have a 4-coloring?Prove that 3-Color ≤P 4-Color.(hint: add vertex to 3-Color problem instance.)

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…

Given an undirected graph G = <V,E>, a vertex cover is a subset of vertices S  V such that for each edge (u,v) belongs to E, either u  S or v  S or both. The Vertex Cover Problem is to find minimum size of the set S. Consider the following algorithm to Vertex Cover Problem: (1) Initialize the result as {} (2) Consider a set of all edges in given graph. Let the set be E’. (3) Do following while E’ is not empty ...a) Pick an arbitrary edge (u,v) from set E’ and add u and v to result ...b) Remove all edges from E which are either incident on u or v. (4) Return result. It claim that this algorithm is exact for undirected connected graphs. Is this claim True or False? Justify the answer.