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7. Prove that for every tournament T, there exists some vertex s for which there is...
Say that we have an undirected graph G(V, E) and a pair of vertices s, t and a vertex v that we call a a desired middle vertex . We wish to find out if there exists a simple path (every vertex appears at most once) from s to t that goes via v. Create a flow network by making v a source. Add a new vertex Z as a sink. Join s, t with two directed edges of capacity...
A tournament T is a directed graph G = (V,A), with vertex set V and arc set A, such that for every u,v V, u ≠ v, either (u,v) A or (v,u) A, but not both. Draw a tournament graph that has six vertices.
Question+ Let T be a tree. Prove, direct from the definition of tree, that: (a) Every edge of T is a bridge. Hint: If an edge e a,b E E(T) is not a bridge, is there a path from a to b that avoids e? Why? What does this imply about circuits? (b) Every vertex of T with degree more than 1 is a cut vertex. Hint: If E V(T) has degree 2 or more there must be a path...
Prove that in every simple graph there is a path from every vertex of odd degree to a vertex of odd degree.
Exercise 5. Let G be a graph in which every vertex has degree at least m. Prove that there is a simple path (i.e. no repeated vertices) in G of length m.
Prove that, if G is a nontrivial connected graph in which every vertex has even degree, then the edge connectivity of G is no less than 2. Prove that, if G is a nontrivial connected graph in which every vertex has even degree, then the edge connectivity of G is no less than 2.
1. Consider a directed graph with distinct and non-negative edge lengths and a source vertex s. Fix a destination vertex t, and assume that the graph contains at least one s-t path. Which of the following statements are true? [Check all that apply.] ( )The shortest (i.e., minimum-length) s-t path might have as many as n−1 edges, where n is the number of vertices. ( )There is a shortest s-t path with no repeated vertices (i.e., a "simple" or "loopless"...
Given a directed graph with positive edge lengths and a specified vertex v in the graph, the "all-pairs" v-constrained shortest path problem" is the problem of computing for each pair of vertices i and j the shortest path from i to j that goes through the vertex v. If no such path exists, the answer is . Describe an algorithm that takes a graph G= (V; E) and vertex v as input parameters and computes values L(i; j) that represent...
a. (15 marks) i (7 marks) Consider the weighted directed graph below. Carry out the steps of Dijkstra's shortest path algorithm as covered in lectures, starting at vertex S. Consequently give the shortest path from S to vertex T and its length 6 A 2 3 4 S T F ii (2 marks) For a graph G = (V, E), what is the worst-case time complexity of the version of Dijkstra's shortest path algorithm examined in lectures? (Your answer should...
10. You are given a directed graph G(V, E) where every vertex vi E V is associated with a weight wi> 0. The length of a path is the sum of weights of all vertices along this path. Given s,t e V, suggest an O((n+ m) log n) time algorithm for finding the shortest path m s toO As usual, n = IVI and m = IEI.