Cut edge vs cut vertices -CONFUSION

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In summary, a cut edge is an edge that, when removed, increases the number of connected components in a graph, while a cut vertex is a vertex that, when removed, does the same. Both cut edges and cut vertices are used to determine the connectivity of a graph, and can be identified by removing them and observing how many connected components are left.
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yakin
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How to find if a graph has a cut-edge/bridge and cut vertices. In other words, i am mixed up between cut edge and cut vertices? What is the difference and how one can find these in a graph?
 
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yakin said:
How to find if a graph has a cut-edge/bridge and cut vertices. In other words, i am mixed up between cut edge and cut vertices? What is the difference and how one can find these in a graph?

Hi yakin, :)

Cut edges are edges when deleted increases the number of connected components in the graph. That is by deleting a cut edge you will disconnect the graph into $n+k$ connected components where $n$ is the current number of connected components and $k$ is an integer which depends on the cut edge you remove. Similarly a cut vertex is a vertex when deleted increases the number of connected components in the graph. Here is a picture depicting cut edges and cut vertices. The cut edges are marked in red and the cut vertices are marked in black.

23nxw4.png
 

FAQ: Cut edge vs cut vertices -CONFUSION

What is the difference between a cut edge and a cut vertex?

A cut edge is an edge in a graph that, when removed, disconnects the graph into two or more separate components. A cut vertex, on the other hand, is a vertex in a graph that, when removed, disconnects the graph into two or more separate components.

How do cut edges and cut vertices affect the connectivity of a graph?

Cut edges and cut vertices both play a crucial role in determining the connectivity of a graph. Removing a cut edge or cut vertex can result in the graph being disconnected, while adding a cut edge or cut vertex can increase the connectivity of the graph.

Can a cut edge also be a cut vertex?

Yes, it is possible for an edge to be both a cut edge and a cut vertex. This occurs when the edge connects two components of the graph and its removal results in the graph being disconnected into more than two components.

How can I identify cut edges and cut vertices in a graph?

To identify a cut edge, you can use the concept of a bridge - an edge whose removal disconnects the graph. Cut vertices can be identified by using the concept of an articulation point - a vertex whose removal disconnects the graph.

Are cut edges and cut vertices important in graph theory?

Yes, cut edges and cut vertices are fundamental concepts in graph theory. They help us understand the connectivity of a graph and can be used to solve various graph-related problems, such as finding the shortest path between two vertices or determining the robustness of a network.

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