# Constructing a graph with radius two.

From cycles $$C_n$$, $$n\geq6$$, I was trying to form a new graph by adding a single vertex to $$C_n$$ so that the added vertex has eccentricity two and rest have three. I tried for $$C_6$$ and $$C_7$$ as given. But unable to do the same for $$C_8$$. Kindly suggest me how to proceed. Any hint will be appreciated. Thanks a lot for the help.

P.S. In the figures, red vertex has the eccentricity two.

• Do you have some reason to think that's possible? Feb 18, 2019 at 12:50

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• Further, I think it can not be extended for $C_{10}$ and higher classes of cycles. I mean cannot be done by adding one vertex. Am I right? Feb 18, 2019 at 14:16
• Yes, you are right. For $C_9$ it can still be done, but not for larger cycles. Feb 18, 2019 at 15:09
• Not sure what you are asking: do you want both new vertices to have eccentricity $2$ and all old vertices eccentricity $3$ and check if this is possible for all cycles? I have not thought about this. Feb 19, 2019 at 10:10