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Please complete ALL parts with neat handwriting and ignore references to tutorial and drills. Please let me know if more clarification is needed.

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[1] (16 pts.) Independent Set and Clique, Alternate BFFs In Tutorial 7 you show that a graph G has an independent set of size k if and only if G has a vertex cover of size n - k. At a high level, this tells us that if we know how to find the largest independent set S, then we can easily construct the smallest vertex cover (V - S) and vice-versa. The goal of this problem will to be to prove this kind of correspondence between independent sets and maximum cliques, explored below (and in Drill 16). In an undirected graph G = (V, E), a subset of the nodes C CV is a clique of size k if 1. |C| = k (there are k nodes in C), and 2. Vu, v E C:{u, v} E (all pairs of nodes from C have an edge connecting them.) This problem explores the cliques and their relation to independent sets. a. (1 pt.) Tutorial 7 determines the maximum size of an independent set in a cycle graph Cn on n vertices. What is the maximum size of a clique in the cycle graph Cn for n > 3? b. (1 pt.) Tutorial 7 determines the maximum size of an independent set in a bipartite graph with left and right "sides L and R. What is the maximum size of a clique in this graph? Given a graph G = (V, E), we define its negative graph G = (V,) such that for all u,v e V, {u, v} E E = {u, v} & . (Intuitively, G has edges precisely where G does not, and vice-versa.) C. (2 pts.) If G has n nodes and m edges, how many nodes and edges does G have? Explain. d. (3 pts.) What is the maximum size of a clique in the negative cycle graph Cn for n > 3? Explain. e. (3 pts.) What is the maximum size of a clique in the negative bipartite graph G = (LUR, E), in terms of L,R? Explain. - f. (6 pts.) Prove that a graph G = (V, E) has an independent set S of size k if and only if the graph G = (V, ) has a clique C of size k. 1 Question 1 Wherein we describe the point distribution of the expected solution. Obviously this will not cover everyone but I can't write a rubric for all possible solutions that I haven't even seen yet. 1a. 1 pt. Correctly determine max clique size in Cn. 1b. 1 pt. Correctly determine max clique size in a bipartite graph. 1c. 1 pt. Correctly determine the number of edges in the negative graph. 1 pt. Provide a brief explanation. 1d. 1 pt. Correctly determine the number of edges in the negative cycle graph. 2 pt. Provide a brief explanation. le. 1 pt. Correctly determine the number of edges in the negative bipartite graph. 2 pt. Provide a brief explanation. le. 2 pt. Proof must somehow address both directions of the if-and-only-if. 3 pt. Actual proof must be correct and use appropriate definitions, algebra, etc, without skipping too many steps. 1 pt. Must have both (word) explanations and formal statements in predicate logic (or its equivalent.) [1] (16 pts.) Independent Set and Clique, Alternate BFFs In Tutorial 7 you show that a graph G has an independent set of size k if and only if G has a vertex cover of size n - k. At a high level, this tells us that if we know how to find the largest independent set S, then we can easily construct the smallest vertex cover (V - S) and vice-versa. The goal of this problem will to be to prove this kind of correspondence between independent sets and maximum cliques, explored below (and in Drill 16). In an undirected graph G = (V, E), a subset of the nodes C CV is a clique of size k if 1. |C| = k (there are k nodes in C), and 2. Vu, v E C:{u, v} E (all pairs of nodes from C have an edge connecting them.) This problem explores the cliques and their relation to independent sets. a. (1 pt.) Tutorial 7 determines the maximum size of an independent set in a cycle graph Cn on n vertices. What is the maximum size of a clique in the cycle graph Cn for n > 3? b. (1 pt.) Tutorial 7 determines the maximum size of an independent set in a bipartite graph with left and right "sides L and R. What is the maximum size of a clique in this graph? Given a graph G = (V, E), we define its negative graph G = (V,) such that for all u,v e V, {u, v} E E = {u, v} & . (Intuitively, G has edges precisely where G does not, and vice-versa.) C. (2 pts.) If G has n nodes and m edges, how many nodes and edges does G have? Explain. d. (3 pts.) What is the maximum size of a clique in the negative cycle graph Cn for n > 3? Explain. e. (3 pts.) What is the maximum size of a clique in the negative bipartite graph G = (LUR, E), in terms of L,R? Explain. - f. (6 pts.) Prove that a graph G = (V, E) has an independent set S of size k if and only if the graph G = (V, ) has a clique C of size k. 1 Question 1 Wherein we describe the point distribution of the expected solution. Obviously this will not cover everyone but I can't write a rubric for all possible solutions that I haven't even seen yet. 1a. 1 pt. Correctly determine max clique size in Cn. 1b. 1 pt. Correctly determine max clique size in a bipartite graph. 1c. 1 pt. Correctly determine the number of edges in the negative graph. 1 pt. Provide a brief explanation. 1d. 1 pt. Correctly determine the number of edges in the negative cycle graph. 2 pt. Provide a brief explanation. le. 1 pt. Correctly determine the number of edges in the negative bipartite graph. 2 pt. Provide a brief explanation. le. 2 pt. Proof must somehow address both directions of the if-and-only-if. 3 pt. Actual proof must be correct and use appropriate definitions, algebra, etc, without skipping too many steps. 1 pt. Must have both (word) explanations and formal statements in predicate logic (or its equivalent.)

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