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Let G be a connected undirected weighted graph. Consider the following two statements. S1: There exists a minimum weight edge in G which is present in every minimum spanning tree of G. S2: If every edge in G has distinct weight, then G has a unique minimum spanning tree. Which one of the following options is correct?

Let G be a connected undirected weighted graph. Consider the following two statements. S1: There exists a minimum weight edge in G which is present in every minimum spanning tree of G. S2: If every edge in G has distinct weight, then G has a unique minimum spanning tree. Which one of the following options is correct? Correct Answer S<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">1</span> is false and S<span style="font-size: 10.5px;">2</span> is true.

Concept:

A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges(V – 1 ) of a connected, edge-weighted undirected graph G(V, E) that connects all the vertices together, without any cycles and with the minimum possible total edge weight.

Statement I: Incorrect

Diagram: change weight and  remove directions and weight is 1:

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[ alt="F1 Raju.s 26-02-21 Savita D2" src="//storage.googleapis.com/tb-img/production/21/02/F1_Raju.s_26-02-21_Savita_D2.png" style="width: 506px; height: 174px;">

Since edge(AB) = edge(AC) = edge(BC) = minimum weight edge

While forming MST: not a single edge is present in all MST

Statement II: Correct:

If every edge in G has distinct weight, then G has a unique minimum spanning tree

Graph G(V, E)

[ alt="Assignment 3 Neetu GATE 2017 Set 1 21 - 30 10 Qs 1&2Marks Solution Modified images Raju D 2" src="//storage.googleapis.com/tb-img/production/19/12/Assignment%203_Neetu_GATE%202017%20Set%201_21%20-%2030_10%20Qs_1%262Marks_Solution_Modified_images_Raju_D%202.PNG" style="width: 148px; height: 114px;">

G(V, V – 1) → minimum spanning tree

[ alt="Assignment 3 Neetu GATE 2017 Set 1 21 - 30 10 Qs 1&2Marks Solution Modified images Raju D 3" src="//storage.googleapis.com/tb-img/production/19/12/Assignment%203_Neetu_GATE%202017%20Set%201_21%20-%2030_10%20Qs_1%262Marks_Solution_Modified_images_Raju_D%203.PNG">

If edge weights are distinct then there exist unique MST.

Hence Statement II is correct.

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