Please complete ALL parts with neat handwriting and ignore references to tutorial and drills. Please let me know if more clarification is needed.
Below is the parts needed for each question. Please complete ALL of what is detailed in the image below.
[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.)
