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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.

enter image description here

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

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    $\begingroup$ Do you have some reason to think that's possible? $\endgroup$
    – Todor Markov
    Feb 18, 2019 at 12:50

1 Answer 1

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enter image description here $$\qquad\qquad\qquad\qquad\qquad$$

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  • $\begingroup$ Thanks a lot. I did some silly mistakes. Thank you very much $\endgroup$
    – monalisa
    Feb 18, 2019 at 13:34
  • $\begingroup$ 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? $\endgroup$
    – monalisa
    Feb 18, 2019 at 14:16
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    $\begingroup$ Yes, you are right. For $C_9$ it can still be done, but not for larger cycles. $\endgroup$
    – Leen Droogendijk
    Feb 18, 2019 at 15:09
  • $\begingroup$ @LeenDroogendijk Means we need to add two vertices for that. Am I right? $\endgroup$
    – monalisa
    Feb 18, 2019 at 19:07
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    $\begingroup$ 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. $\endgroup$
    – Leen Droogendijk
    Feb 19, 2019 at 10:10

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