Given a connected graph. How can that we can always find some vertexes that do not contain any lines between them, remove all the lines between the remaining vertexes and get a connected graph.
Delete some vertexes of a connected graph and it still remains connected
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graph-theory
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0I don't think I understand what you are describing. If you choose a set of vertices that has no edge between these vertices and remove all edges between the other vertices you'd have a graph with no edges – 2017-02-10
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0@user160738 That's not true, in general, but I believe what the OP asks is not true either. – 2017-02-10
1 Answers
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You can't. Consider a cycle with 4 or more vertices and delete any two vertices on that cycle that aren't adjacent. You end up with two connected components, but not a connected graph.
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0This works, but I just want to point out the cycle needs to be large enough ($\geq 6$ vertices). – 2017-02-10
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0Technically you can consider a single vertex a connected component, so your cycle only needs at least 4 vertices. I have edited my answer accordingly. – 2017-02-10
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0I think we can choose what points to select here, no? – 2017-02-10
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0If we can choose a single vertex, yes, you are correct. – 2017-02-10
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0I mean that the points are not randomly chosen. I have rewritten it a bit, maybe this statement will be true? – 2017-02-10
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0Every connected graph $G$ has a connectivity number $\kappa(G)$. If $\kappa(G)=k$ then there exists a set of $k$ vertices that disconnects $G$ and no $k-1$ set that disconnects $G$. Graph connectivity problems can be difficult in general. – 2017-02-10