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Problem 5. (12 marks) Connectivity in undirected graphs vs. directed graphs. a. (8 marks) Prove that...
Look up the definition of a biconnected undirected graph on Wikipedia. Give a one sentence definition based on induced sub-graphs. Start your definition with “An undirected graph G = (V, E) is biconne...
Look up the definition of a biconnected undirected graph on
Wikipedia. Give a one sentence definition based on induced
sub-graphs. Start your definition with “An undirected graph G = (V,
E) is biconnected, if . . . ” (b) For a directed graph G = (V, E),
its underlying undirected graph is obtained by replacing every
directed edge (u, v) with an undirected one {u, v}. (If (u, v) and
(v, u) are both in E, then the underlying undirected...
7. Graphs u, u2, u3, u4, u5, u6} and the (a) Consider the undirected graph G...
7. Graphs u, u2, u3, u4, u5, u6} and the (a) Consider the undirected graph G (V, E), with vertex set V set of edges E ((ul,u2), (u2,u3), (u3, u4), (u4, u5), (u5, u6). (u6, ul)} i. Draw a graphical representation of G. ii. Write the adjacency matrix of the graph G ii. Is the graph G isomorphic to any member of K, C, Wn or Q? Justify your answer. a. (1 Mark) (2 Marks) (2 Marks) b. Consider an...
Write down true (T) or false (F) for each statement. Statements are shown below If a...
Write down true (T) or false (F) for each statement. Statements are shown below If a graph with n vertices is connected, then it must have at least n − 1 edges. If a graph with n vertices has at least n − 1 edges, then it must be connected. If a simple undirected graph with n vertices has at least n edges, then it must contain a cycle. If a graph with n vertices contain a cycle, then it...
1. You will be asked questions about graphs. The graphs are provided formally. To answers the...
1. You will be asked questions about graphs. The graphs are provided formally. To answers the questions, it may help to draw the graphs on a separate sheet. a Consider the graph (V, E), V = {a,b,c,d) and E = {{a,d}, {b,d}, {c, d}}. This graph is directed/undirected This graph is a tree y/n. If yes, the leafs are: This graph is bipartite y/n. If yes, the partitions are: a, d, b, c is/is not a path in this graph....
Problem 5. (15 marks) Given a connected, undirected, weighted graph G- (V, E), define the cost...
Problem 5. (15 marks) Given a connected, undirected, weighted graph G- (V, E), define the cost of a spanning tree to be the maximum weight among the weights associated with the edges of the spanning tree. Design an efficient algorithm to find the spanning tree of G which maximize above defined cost What is the complexity of your algorithm.
Remarks: All the graphs here are without self loops and parallel edges, and anti-parallel edges. When...
Remarks: All the graphs here are without self loops and parallel edges, and anti-parallel edges. When we speak of a flow network, we mean there are capacities c(e) ? 0 on the edges, the graph G is directed with a source s and a destination t. In all the algorithms, always explain their correctness and analyze their complexity. The complexity should be as small as possible. A correct algorithm with large complexity, may not get full credit. • Question 3:...
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more tha...
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more than one edge between the same pair of vertices) or self-loops (edges with both ends at the same vertex). [10 points] a. A graph with 3 connected components, 11 vertices, and 10 edges. b. A graph with 4 connected components, 10 vertices, and 30 edges. c....
Question 1: Given an undirected connected graph so that every edge belongs to at least one...
Question 1: Given an undirected connected graph so that every edge belongs to at least one simple cycle (a cycle is simple if be vertex appears more than once). Show that we can give a direction to every edge so that the graph will be strongly connected. Question 2: Given a graph G(V, E) a set I is an independent set if for every uv el, u #v, uv & E. A Matching is a collection of edges {ei} so...
2. Let G = (V, E) be an undirected connected graph with n vertices and with...
2. Let G = (V, E) be an undirected connected graph with n vertices and with an edge-weight function w : E → Z. An edge (u, v) ∈ E is sparkling if it is contained in some minimum spanning tree (MST) of G. The computational problem is to return the set of all sparkling edges in E. Describe an efficient algorithm for this computational problem. You do not need to use pseudocode. What is the asymptotic time complexity of...
I have done the a and b, but i'm so confuse with other questions, could someone help me to fix these questions, thanks so much. 4 Directed graphs Directed graphs are sometimes used operating syst...
I have done the a and b, but i'm so confuse with other
questions, could someone help me to fix these questions, thanks so
much.
4 Directed graphs Directed graphs are sometimes used operating systems when trying to avoid deadlock, which is a condition when several processes are waiting for a resource to become available, but this wil never happen because Page 2 p2 T2 Figure 1: Minimal example of a resource allocation graph with deadlock other processes are holding...
Look up the definition of a biconnected undirected graph on
Wikipedia. Give a one sentence definition based on induced
sub-graphs. Start your definition with “An undirected graph G = (V,
E) is biconnected, if . . . ” (b) For a directed graph G = (V, E),
its underlying undirected graph is obtained by replacing every
directed edge (u, v) with an undirected one {u, v}. (If (u, v) and
(v, u) are both in E, then the underlying undirected...
7. Graphs u, u2, u3, u4, u5, u6} and the (a) Consider the undirected graph G (V, E), with vertex set V set of edges E ((ul,u2), (u2,u3), (u3, u4), (u4, u5), (u5, u6). (u6, ul)} i. Draw a graphical representation of G. ii. Write the adjacency matrix of the graph G ii. Is the graph G isomorphic to any member of K, C, Wn or Q? Justify your answer. a. (1 Mark) (2 Marks) (2 Marks) b. Consider an...
1. You will be asked questions about graphs. The graphs are provided formally. To answers the questions, it may help to draw the graphs on a separate sheet. a Consider the graph (V, E), V = {a,b,c,d) and E = {{a,d}, {b,d}, {c, d}}. This graph is directed/undirected This graph is a tree y/n. If yes, the leafs are: This graph is bipartite y/n. If yes, the partitions are: a, d, b, c is/is not a path in this graph....
Problem 5. (15 marks) Given a connected, undirected, weighted graph G- (V, E), define the cost of a spanning tree to be the maximum weight among the weights associated with the edges of the spanning tree. Design an efficient algorithm to find the spanning tree of G which maximize above defined cost What is the complexity of your algorithm.
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more than one edge between the same pair of vertices) or self-loops (edges with both ends at the same vertex). [10 points] a. A graph with 3 connected components, 11 vertices, and 10 edges. b. A graph with 4 connected components, 10 vertices, and 30 edges. c....
I have done the a and b, but i'm so confuse with other
questions, could someone help me to fix these questions, thanks so
much.
4 Directed graphs Directed graphs are sometimes used operating systems when trying to avoid deadlock, which is a condition when several processes are waiting for a resource to become available, but this wil never happen because Page 2 p2 T2 Figure 1: Minimal example of a resource allocation graph with deadlock other processes are holding...
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