1.Consider a complete graph G with 4 vertices. The graph G has. ---- spanning trees. 2. Consider a undirected graph G with vertices { A, B, C, D, EI. In graph 6, every edge has distinet weight. Edge CD is edge with minimum weight and edge AB is edge with maximum weight. Then, which of the following is false? a) Every minimum spanning tree of G must contain CD b) IF AB is in a minimum spanning tree, then its removal must disconnect G c) No minimum spanning tree contains AB d) G has a unique minimum spanning tree

1.Consider a complete graph G with 4 vertices. The graph G has. ---- spanning trees. 2. Consider a undirected graph G with vertices { A, B, C, D, EI. In graph 6, every edge has distinet weight. Edge CD is edge with minimum weight and edge AB is edge with maximum weight. Then, which of the following is false? a) Every minimum spanning tree of G must contain CD b) IF AB is in a minimum spanning tree, then its removal must disconnect G c) No minimum spanning tree contains AB d) G has a unique minimum spanning tree

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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let us take any standard graph G=(v,e) and let us pretend each edge is the same exact weight. let us think about a minimum spanning tree of the graph G, called T = (V, E' ). under each part a and b illustrate then show that a) s a unique path between u and v in T for all u, v ∈ V . b) tree T is not unique. provide proof
G = (V,E,W) is a weighted connected (undirected) graph where all edges have distinct weights except two edges e and e′ which have the same weight. Suppose there is a Minimum Spanning Tree of G containing both e and e′. Prove that G has a unique Minimum Spanning Tree.
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. Let G be a weighted, connected, undirected graph, and let V1 and V2 be a partition of the vertices of G into two disjoint nonempty sets. Furthermore, let e be an edge in the minimum spanning tree for G such that e has one endpoint in V1 and the other in V2. Give an example that shows that e is not necessarily the smallest- weight edge that has one endpoint in V1 and the other in V2.
Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Prove by contradiction or counterexample. Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.
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