An orientation of an undirected graph G = (V, E) is an assignment of a direction...
An orientation of an undirected graph G = (V, E) is an assignment of a direction to each edge e ∈ E. An acyclic orientation is the assignment of a direction to every edge such that the resulting directed graph contains no cycles. Either prove that there exist undirected graphs with no acyclic orientation, or provide an efficient O(V +E) algorithm for producing an acyclic orientation for an undirected graph G and explain why it produces a valid acyclic orientation.
Homework Answers
.Its a long proof and please download the attachment to download the proof :
https://drive.google.com/open?id=1n5spVIIaZk-pY1o-kT6mXRz5ICSzZeet
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An orientation of an undirected graph G = (V, E) is an
assignment of a direction...
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...
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...
Problem 5. (12 marks) Connectivity in undirected graphs vs. directed graphs. a. (8 marks) Prove that...
Problem 5. (12 marks) Connectivity in undirected graphs vs. directed graphs. a. (8 marks) Prove that in any connected undirected graph G- (V, E) with VI > 2, there are at least two vertices u, u є V whose removal (along with all the edges that touch them) leaves G still connected. Propose an efficient algorithm to find two such vertices. (Hint: The algorithm should be based on the proof or the proof should be based on the algorithm.) b....
Let G = (V, E) be a weighted undirected connected graph that contains a cycle. Let...
Let G = (V, E) be a weighted undirected connected graph that contains a cycle. Let k ∈ E be the edge with maximum weight among all edges in the cycle. Prove that G has a minimum spanning tree NOT including k.
2) Let G ME) be an undirected Graph. A node cover of G is a subset...
2) Let G ME) be an undirected Graph. A node cover of G is a subset U of the vertex set V such that every edge in E is incident to at least one vertex in U. A minimum node cover MNC) is one with the lowest number of vertices. For example {1,3,5,6is a node cover for the following graph, but 2,3,5} is a min node cover Consider the following Greedy algorithm for this problem: Algorithm NodeCover (V,E) Uempty While...
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?...
Let G = (V. E) be an undirected, connected graph with weight function w : E → R
Problem 4 Let G = (V. E) be an undirected, connected graph with weight function w : E → R. Furthermore, suppose that E 2 |V and that all edge weights are distinct. Prove that the MST of G is unique (that is, that there is only one minimum spanning tree of G).
B-1 Graph coloring Given an undirected graph G (V. E), a k-coloring of G is a function c : V → {0, 1, . . . ,k-1}
B-1 Graph coloring Given an undirected graph G (V. E), a k-coloring of G is a function c : V → {0, 1, . . . ,k-1} such that c(u)≠c(v) for every edge (u, v) ∈ E. In other words, the numbers 0.1,... k-1 represent the k colors, and adjacent vertices different colors. must havec. Let d be the maximum degree of any vertex in a graph G. Prove that we can color G with d +1 colors.
You are given an undirected graph G = (V, E) with positive weights on the edges....
You are given an undirected graph G = (V, E) with positive weights on the edges. If the edge weights are distinct, then there is only one MST, so both Prim’s and Kruskal’s algorithms will find the same MST. If some of the edge weights are the same, then there can be several MSTs and the two algorithms could find different MSTs. Describe a method that forces Prim’s algorithm to find the same MST of G that Kruskal’s algorithm finds.
Let G = (V;E) be an undirected and unweighted graph. Let S be a subset of the vertices. The graph...
Let G = (V;E) be an undirected and unweighted graph. Let S be a subset of the vertices. The graph induced on S, denoted G[S] is a graph that has vertex set S and an edge between two vertices u, v that is an element of S provided that {u,v} is an edge of G. A subset K of V is called a killer set of G if the deletion of K kills all the edges of G, that is...
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...
Problem 5. (12 marks) Connectivity in undirected graphs vs. directed graphs. a. (8 marks) Prove that in any connected undirected graph G- (V, E) with VI > 2, there are at least two vertices u, u є V whose removal (along with all the edges that touch them) leaves G still connected. Propose an efficient algorithm to find two such vertices. (Hint: The algorithm should be based on the proof or the proof should be based on the algorithm.) b....
2) Let G ME) be an undirected Graph. A node cover of G is a subset U of the vertex set V such that every edge in E is incident to at least one vertex in U. A minimum node cover MNC) is one with the lowest number of vertices. For example {1,3,5,6is a node cover for the following graph, but 2,3,5} is a min node cover Consider the following Greedy algorithm for this problem: Algorithm NodeCover (V,E) Uempty While...
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?...
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