Homework Answers
For A disconnected graph with n nodes, take n-1 nodes and make it fully connected with (n-1)x(n-2) /2 edges and leave the last nth node as it is.
So, the maximum value of E is (n-1) (n-2) / 2
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Problem 6. In lecture, we saw that an undirected graph with n nodes can have at...
A graph with n nodes is connected, undirected, and acyclic. How many edges must it have? (Select ...
A graph with n nodes is connected, undirected, and acyclic. How many edges must it have? (Select the answer from the following options and prove your choice): a) n b) n*(n-1) c) n- 1 d) n/2 - 1
We now consider undirected graphs. Recall that such a graph is • connected iff for all...
We now consider undirected graphs. Recall that such a graph is • connected iff for all pairs of nodes u, w, there is a path of edges between u and w; • acyclic iff for all pairs of nodes u, w, whenever there is an edge between u and w then there is no path Given an acyclic undirected graph G with n nodes (where n ≥ 1) and a edges, your task is to prove that a ≤ n...
An undirected bipartite graph has n vertices and m edges. a) If the graph is connected,...
An undirected bipartite graph has n vertices and m edges. a) If the graph is connected, what is the minimum number of edges? b) If the graph is disconnected, what is the maximum number of edges? c) What is the longest single path? d) If the path can pass through a vertex and not any edges more than once, What is the longest path? Kindly provide me with an example for me to relate
Other answer is incorrect Problem 1. (15 points) Consider an undirected connected graph G = (V,...
Other answer is incorrect
Problem 1. (15 points) Consider an undirected connected graph G = (V, E) with edge costs ce > 0 for e € E which are all distinct. (a) [8 points). Let E' CE be defined as the following set of edges: for each node v, E' contains the cheapest of all edges incident on v, i.e., the cheapest edge that has v as one of its endpoints. Is the graph (V, E') connected? Is it acyclic?...
Have the explaination please. 4 Graph Application: Network Connectivity (Adapted from Problem 9, Chapter 3 of...
Have the explaination please.
4 Graph Application: Network Connectivity (Adapted from Problem 9, Chapter 3 of K&T) Think of a communications network as a connected, undi rected graph, where messages from one node s to another node t are sent along paths from s to t. Nodes can sometimes fail. If a node v fails then no messages can be sent along edges incident on v. A network is particularly vulnerable if failure of a single node v can cause...
There are n bus stations, they are connected by m roads (there is a undirected graph...
There are n bus stations, they are connected by m roads (there is a undirected graph with n vertexes and m edges) The cost for advertising at bus station i is Ci. Now, give you two bus stations s, t, design an algorithm: select some bus stations, advertise at these stations, and every route from s to t can go through at least one advertising bus station. Minimize the total cost. Try to reduce it as a max s-t flow...
Define the graph Gn to have the 2n nodes 20, 21,...,an-1, bo, b1, ..., bn-1 and...
Define the graph Gn to have the 2n nodes 20, 21,...,an-1, bo, b1, ..., bn-1 and the following edges. Each node ai, for i = 0,1,...,n - 1, is connected to the nodes b; and bk, where j = 2i mod n and k = (2i + 1) mod n (a) Prove that for every n, G, has a perfect matching. (b) How many different perfect matchings does G100 have?
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...
Problem 2: As we discussed in class, one can use an algorithm for computing all-pairs shortest...
Problem 2: As we discussed in class, one can use an algorithm for computing all-pairs shortest paths to also compute the transitive closure of a graph. If using Floyd-Warshall for example, it is possible to do this in On") time (where as usual n is the number of nodes and m is the number of edges). Show how to compute the transitive closure of a directed graph in O(nm) time. For which type of graphs is this better than using...
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...
Other answer is incorrect
Problem 1. (15 points) Consider an undirected connected graph G = (V, E) with edge costs ce > 0 for e € E which are all distinct. (a) [8 points). Let E' CE be defined as the following set of edges: for each node v, E' contains the cheapest of all edges incident on v, i.e., the cheapest edge that has v as one of its endpoints. Is the graph (V, E') connected? Is it acyclic?...
Have the explaination please.
4 Graph Application: Network Connectivity (Adapted from Problem 9, Chapter 3 of K&T) Think of a communications network as a connected, undi rected graph, where messages from one node s to another node t are sent along paths from s to t. Nodes can sometimes fail. If a node v fails then no messages can be sent along edges incident on v. A network is particularly vulnerable if failure of a single node v can cause...
Define the graph Gn to have the 2n nodes 20, 21,...,an-1, bo, b1, ..., bn-1 and the following edges. Each node ai, for i = 0,1,...,n - 1, is connected to the nodes b; and bk, where j = 2i mod n and k = (2i + 1) mod n (a) Prove that for every n, G, has a perfect matching. (b) How many different perfect matchings does G100 have?
Problem 2: As we discussed in class, one can use an algorithm for computing all-pairs shortest paths to also compute the transitive closure of a graph. If using Floyd-Warshall for example, it is possible to do this in On") time (where as usual n is the number of nodes and m is the number of edges). Show how to compute the transitive closure of a directed graph in O(nm) time. For which type of graphs is this better than using...
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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