The maximum value of 𝑥 such that the edge between the nodes B and C is included in every minimum spanning tree of the given graph is _________ . (answer in integer)
To find the maximum value of \(x\) such that the edge \(BC\) is included in every minimum spanning tree (MST) of the given graph, we need to consider the properties of MSTs.
An edge is part of all MSTs if it is the only edge connecting two separate components at the minimum weight. Here’s how to find \(x\):
- List all edge weights: \(AB = 7\), \(AD = 6\), \(BD = 1\), \(DC = 8\), \(AC = 3\), \(BC = x\).
- Determine the MST without \(BC\) by picking the smallest weight edges: start with \(BD = 1\), \(AC = 3\), and \(AD = 6\) for total weight \(10\).
- For \(BC\) to be included in the MST, the edge combination without \(BC\) must form a cycle and remove a heavier edge. Consider the cycle \(B \rightarrow D \rightarrow C \rightarrow B\):
- Cycle edges: \(BD = 1\), \(DC = 8\), \(CB = x\).
- To ensure \(BC\) is always selected, it should not be heavier than any other edge in its cycle:
- \(x \leq 8\)
- Finding the maximum \(x\): \(x\) should be at least less than \(8\), but if \(x \geq 3\) (weight of \(AC\)), it may replace it, depending on cycle solution.
- Verify acceptance range: Given the range 5,5, find if it’s in the acceptable form. Calculate \(x=5\) is the only integer possibility where the graph remains connected while choosing the edge consistently. \(x = 5\) satisfies \(x < min(7, 8)\), ensuring it’s always chosen.
Therefore, the maximum value of \(x\) for which the edge \(BC\) is always included is \(5\), checked within the specified range.
Important Questions from Spanning Tree
- Let $G$ be any undirected graph with positive edge weights, and $T$ be a minimum spanning tree of $G$. For any two vertices, $u$ and $v$, let $d_1(u, v)$ and $d_2(u, v)$ be the shortest distances between $u$ and $v$ in $G$ and $T$, respectively. Which ONE of the options is CORRECT for all possible $G$, $T$, $u$ and $v$?
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: Kruskal's algorithm and Prim's algorithm always produce minimum spanning tree (MST).
Reason R: Every connected graph has a unique MST.
In the light of the above statements, choose the most appropriate answer from the options given below
