[Graph Theory - Discrete Mathematics] How do you solve this? Let G=(V,E) be a co

CMIIh 2021-08-16 Answered
[Graph Theory - Discrete Mathematics] How do you solve this?
Let \(\displaystyle{G}={\left({V},{E}\right)}\) be a connected simple graph, and call the edge \(\displaystyle{\left({u},{v}\right)}\in{E}\) essential if the graph obtained by removing (u, v) from G is disconnected. Show that G is a tree if and only if all its edges are essential.

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Expert Answer

Sally Cresswell
Answered 2021-08-17 Author has 13315 answers
Step 1
Given
Let \(\displaystyle{G}={\left({V},{E}\right)}\) be a connected simple graph, and call the edge \(\displaystyle{\left({u},{v}\right)}\in{E}\) essential if the graph obtained by removing (u,v) from G is disconnected.
Step 2
Solution
To show that G is a tree if and only if all its edges are connected.
Since, all vertices in a tree are path connected with a unique path in between them.
So removing an edge from the graph will make it disconnected or just as two distinct subgraphs.
Thus, easily conclude that G is a tree if and only if all its edges are essential.
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