Prove that if an edge (u, v) is in a MST for a graph G, then (u, v) is a light edge crossing some cut in G.
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A: the answer is an given below :
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A: The answer is given below:-
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- Show that an MST of an undirected graph is equivalent to abottleneck SPT of the graph: For every pair of vertices v and w, it gives the path connecting them whose longest edge is as short as possible.Show that if all edges of a graph G have pairwise distinct weights, then thereis exactly one MST for G.Let G = (V, E) be an undirected graph with at least two distinct vertices a, b ∈ V . Prove that we can assign a direction to each edge e ∈ E as to form a directed acyclic graph G′ where a is a source and b is a sink.
- If a graph G = (V, E), |V | > 1 has N strongly connected components, and an edge E(u, v) is removed, what are the upper and lower bounds on the number of strongly connected components in the resulting graph? Give an example of each boundary case.Let G be a connected graph that has exactly 4 vertices of odd degree: v1,v2,v3 and v4. Show that there are paths with no repeated edges from v1 to v2, and from v3 to v4, such that every edge in G is in exactly one of these paths.Show that a bottleneck SPT of a graph is identical to an MST of an undirected graph. It provides the path between each pair of vertices v and w whose longest edge is as short as feasible.
- Prove that if G is a connected graph, then there always is a closed walk that passes through each edge at least once and at most twice.Say that a graph G has a path of length three if there exist distinct vertices u, v, w, t with edges (u, v), (v, w), (w, t). Show that a graph G with 99 vertices and no path of length three has at most 99 edges.Give, with justification, a formula for the minimum number of edges that must be added to a general connected graph to make it have an Euler tour, provided we allow multiple edges (that is, two vertices can be joined by more than one edge).
- Consider a graph G that has k vertices and k −2 connected components,for k ≥ 4. What is the maximum possible number of edges in G? Proveyour answer.Prove 1 For a graph G = (V, E), a forest F is any set of edges of G that doesnot contain any cycles. M = (E, F) where F = {F ⊆ E : F is a forest of G} is amatroid.Show that in an undirected graph, classifying an edge .u; / as a tree edge or a back edge according to whether .u; / or .; u/ is encountered first during the depth-first search is equivalent to classifying it according to the ordering of the four types in the classification scheme.