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What are non-separable graphs? To understand non-separable graphs, we need to understand cut vertices. A vertex of a graph is a cut vertex if deleting it disconnects the graph or the component the vertex belongs to.
Here is my lesson on cut vertices: https://www.youtube.com/watch?v=D1nYRgXPRyM
A non-trivial connected graph is non-separable if it contains NO cut vertices, so deleting any vertex from the graph will not disconnect it. We may be able to disconnect the graph by deleting two vertices, but never by deleting just one. This touches on the idea of vertex connectivity, here is my lesson on that: https://www.youtube.com/watch?v=rK0zHB1WnQE
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