What is the name given to the sub-graph in which all vertices are connected to each other i.e., the subgraph is complete graph? The decision problem of this sub-graph falls under which class? Answer Choices: a) Subset sum, NP Hard b) Clique, NP Hard c) Hamiltonian graph, NP Complete d) Clique, NP Complete
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A: ANSWER:-
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A: Note:the solution to the above problem is given below Hope it helps you understand
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- Clique This is to help you understand the proof that Clique is NPC. Consider the 3-CNF problem: Is there an assignment to φ = (¬x1 ∨ x2 ∨ x3) ∧ (x1 ∨ ¬x2 ∨ ¬x3) ∧ (x1 ∨ x2 ∨ x3)) that satisfies φ? Convert this problem into a graph problem of the form : "Here is a graph. Does it have a clique of size 3?" I want to see the graph. Draw it as cleanly as possible.A complete graph has no loops, and each of its vertexes is adjacent to all its other vertexes. Which complete graphs (if any) are bipartite? Which complete graphs (if any) are not bipartite? Prove that your answers are correct. Hints: think about proofs by cases and graph coloring.(a) Let G be a simple undirected graph with 18 vertices and 53 edges such that the degreesof G are only 3 and 7. Suppose there are a vertices of degree 7 and b vertices ofdegree 3. Find a and b. To receive any credit for this problem you must write completesentences, explain all of your work, and not leave out any details. problem 1, continued(b) Recall that a graph G is said to be k-regular if and only if every vertex in G has degreek. Draw all 3-regular simple graphs with 12 edges (mutually non-isomorphic). Hint:there are six of them. To receive credit for this problem, you should explain, as well aspossible, why all of your graphs are mutually non-isomorphic.
- a) Draw a complete graph K5, Explain, what is the sum of degree count of all vertices in the complete graph, K40 , and (b) Draw a complete bipartite graph, K4,6. Explain, how many edges are in the complete bipartite graph, K40,60.It is stated that K6 is not a planar graph (because K5 is one of the subgraphs of K6), but there is a subgraph of K6 which is a planar graph. If G' is a subgraph of K6, then find G' which has the maximum number of vertices and edges. Hint: Answers are expressed in sets and pictures.2. Q1) You are given an undirected connected planar graph. There are 10 vertices and 7 faces in the graph. What is the number of edges in the graph? Note that: A graph is said to be planar if it can be drawn in a plane so that no edge cross. Q2) In how many ways can we pick any number of balls from a pack of three different balls? Q3) The distance between 2 points A and B is 1320Km. Two cars start moving towards each other with 50 Km/hr and 60 Km/hr. After how many hours do they meet?
- f the edge weights of a connected weighted graph are distinct, the graph must have a unique minimum spanning tree". Is this statement true or false, briefly explain why in a few sentences.A graph is ___________ if there exists a path between any two vertices in the graph. a. connected b. inclusive c. comprehensive d. encapsulatedAll of the following statements are false. Provide a counterexample for each one. iii. If it can be shown that there is not a proper 3-coloring of a graph G, then χ(G) = 4. iv. If G is a graph with χ(G) ≤ 4 then G is planar
- When 3 edges are removed from a graph G without removing any vertices the resulting graph G0 is a forest graph with 5 connected components. The numbers of edges in these components are 11,5,7,10 and 16. Find the number of vertices and the number of edges in G.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…Give an example of a graph that has all 3 of the following properties. (Note that you need to give a single graph as the answer.) (i) It is connected (ii) It has one articulation point. (iii) The graph needs at least 4 colors for a valid vertex coloring (iv) The graph does not have a 4-clique (that is, a clique of 4 vertices) as a subgraph.