Connected Components in Graph Theory

In summary, the conversation discusses finding an algorithm to find a vertex u in a given tree T, such that when removed, the size of each connected component in the resulting graph T' is at most half of the original size. The complexity of this algorithm is discussed, with two possible solutions proposed. The first involves using BFS to find the largest distance from a vertex to any other vertex, and selecting the vertices with the smallest Vm as the center of the graph. The second involves removing all vertices with a degree of 1 until only 1 or 2 vertices remain, and selecting any of those as the center. The complexity for both methods is O(V^2).
  • #1
asi123
258
0

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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  • #2
Can you think of a useful number (or pair of numbers, perhaps) you could associate with each vertex?
 
  • #3
Ok, so I did the following:

I woluld first use BFS algorithm (it has to be run for each verticle) for computng shortest distances between all pairs of verticles. Then for each verticle 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.
 
  • #4
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.
 

1. What are connected components in graph theory?

Connected components in graph theory refer to the subgraphs of a graph that are connected, in which every vertex is reachable from every other vertex by traversing edges of the graph. In other words, a connected component is a group of vertices and edges that are all connected to each other, but not necessarily connected to the rest of the graph.

2. How are connected components identified in a graph?

Connected components can be identified by using graph traversal algorithms such as depth-first search or breadth-first search. These algorithms start at a given vertex and explore all connected vertices, marking them as visited. Any unvisited vertices are then considered to be part of a separate connected component and the algorithm is repeated until all vertices have been visited.

3. What is the significance of connected components in graph theory?

Connected components play an important role in understanding the structure and connectivity of a graph. They can help identify clusters or communities within a network, as well as reveal any isolated vertices or disconnected subgraphs. In addition, many algorithms in graph theory rely on the knowledge of connected components to efficiently solve problems.

4. Can a graph have more than one connected component?

Yes, a graph can have any number of connected components. In fact, a disconnected graph is defined as a graph with two or more connected components. This means that there are at least two subgraphs within the graph that are not connected to each other.

5. How can connected components be used to analyze real-world networks?

Connected components can be used to analyze real-world networks by identifying and analyzing the different clusters or communities within the network. This can help to understand the relationships and connections between different entities in the network, as well as identify any isolated or disconnected subgroups. This information can then be used to make predictions, detect patterns, and solve various problems related to the network.

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