Let G = (V, E) be a directed graph, with source s ∈ V, sink t...

Let G = (V, E) be a directed graph, with source s ∈ V, sink t ∈ V, and nonnegative edge capacities {ce}. Give a polynomial-time algorithm to decide whether G has a unique minimum s-t cut (i.e., an s-t of capacity strictly less than that of all other s-t cuts).

Expert Solution

orchestra answered 2 years ago

Given a graph G = (V,E), the source-sink pair (s,t) and capacity of edges {C_e ≥...

Given a graph G = (V,E), the source-sink pair (s,t) and capacity of edges {C_e ≥ 0 | e ∈ E}, design a polynomial-time algorithm to find a set of edges S, such that for every edge e ∈ S, increasing C_e will lead to an increase of max-flow value between s and t. Show the correctness of your algorithm.

You are given a directed graph G(V,E) with n vertices and m edges. Let S be...

You are given a directed graph G(V,E) with n vertices and m edges. Let S be the subset of vertices in G that are able to reach some cycle in G. Design an O(n + m) time algorithm to compute the set S. You can assume that G is given to you in the adjacency-list representation.

If G = (V, E) is a graph and x ∈ V , let G \...

If G = (V, E) is a graph and x ∈ V , let G \ x be the graph whose vertex set is V \ {x} and whose edges are those edges of G that don’t contain x. Show that every connected finite graph G = (V, E) with at least two vertices has at least two vertices x1, x2 ∈ V such that G \ xi is connected.

A DAG is a directed graph that contains no directed cycles. Define G = (V, E)...

A DAG is a directed graph that contains no directed cycles. Define G = (V, E) in which V is the set of all nodes as {v1, v2, ..., vi , ...vn} and E is the set of edges E = {ei,j = (vi , vj ) | vi , vj ∈ V} . A topological order of a directed graph G = (V, E) is an ordering of its nodes as {v1, v2, ..., vi , ...vn} so that...

Let G = (V, E) be a directed acyclic graph modeling a communication network. Each link...

Let G = (V, E) be a directed acyclic graph modeling a communication network. Each link e in E is associated with two parameters, w(e) and d(e), where w(e) is a non-negative number representing the cost of sending a unit-sized packet through e, and d(e) is an integer between 1 and D representing the time (or delay) needed for transmitting a packet through e. Design an algorithm to find a route for sending a packet between a given pair of...

. Provide a weighted directed graph G = (V, E, c) that includes three vertices a,...

. Provide a weighted directed graph G = (V, E, c) that includes three vertices a, b, and c, and for which the maximum-cost simple path P from a to b includes vertex c, but the subpath from a to c is not the maximum-cost path from a to c

# Problem Description Given a directed graph G = (V,E) with edge length l(e) > 0...

# Problem Description Given a directed graph G = (V,E) with edge length l(e) > 0 for any e in E, and a source vertex s. Use Dijkstra’s algorithm to calculate distance(s,v) for all of the vertices v in V. (You can implement your own priority queue or use the build-in function for C++/Python) # Input The graph has `n` vertices and `m` edges. There are m + 1 lines, the first line gives three numbers `n`,`m` and `s`(1 <=...

Let f(t) =t^2−1 and g(t) =e^t. (a) Graph f(g(t)) and g(f(t)). (b) Which is larger,f(g(5)) or...

Let f(t) =t^2−1 and g(t) =e^t. (a) Graph f(g(t)) and g(f(t)). (b) Which is larger,f(g(5)) or g(f(5))? Justify your answer. (c) Which is larger, (f(g(5)))′or g(f(5))′? Justify your answer.

Let G a graph of order 8 with V (G) = {v1, v2, . . ....

Let G a graph of order 8 with V (G) = {v1, v2, . . . , v8} such that deg vi = i for 1 ≤ i ≤ 7. What is deg v8? Justify your answer. Please show all steps thank you

ii. Let G = (V, E) be a tree. Prove G has |V | − 1...

ii. Let G = (V, E) be a tree. Prove G has |V | − 1 edges using strong induction. Hint: In the inductive step, choose an edge (u, v) and partition the set vertices into two subtrees, those that are reachable from u without traversing (u, v) and those that are reachable from v without traversing (u, v). You will have to reason why these subtrees are distinct subgraphs of G. iii. What is the total degree of a...

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