Connected Components in Graph Theory

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Homework Help Overview

The discussion revolves around finding a vertex in a tree such that removing it results in connected components of size at most |V| / 2. The problem is situated within the context of graph theory, specifically focusing on properties of trees and connected components.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants explore different methods for identifying a suitable vertex, including the use of BFS to compute distances and considering the removal of vertices based on their degree.

Discussion Status

Some participants have proposed methods involving BFS and vertex degree considerations, while others are questioning the efficiency of these approaches. There is an ongoing exploration of potential strategies without a clear consensus on the best method.

Contextual Notes

Participants are considering the complexity of their proposed algorithms and the implications of removing vertices with specific properties, such as degree. There is an acknowledgment of the challenge in achieving better than O(V^2) complexity.

asi123
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Homework Statement



Hey guys,

I have this question.

Given a tree T = (V , F), find an algorithm which finds u in V, so in the graph T = (V \ {u} , F) the size of each connected component is |V| / 2 at most. What is the complexity?

Homework Equations





The Attempt at a Solution



I need to split this tree down the middle somehow, any idea?

Thanks a lot.

Assaf
 
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Can you think of a useful number (or pair of numbers, perhaps) you could associate with each vertex?
 
Ok, so I did the following:

I woluld first use BFS algorithm (it has to be run for each vertical) for computng shortest distances between all pairs of verticles. Then for each vertical V find Vm - the largest distance to any other verticles amongs the data retuirned form BFS. Then, the verticles with the smallest Vm are the one in the graph center.

It comes down to O(V^2), any thing better then that?

Thanks a lot.

Assaf.
 
My method doesn't beat O(v^2), but it's easy to describe. Remove all vertices degree 1. Repeat until only 1 or 2 vertices left. Pick any of those.
 

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