@Jim Can we use this room for discussion?

@ThomasKlimpel of course , your wish is my command, but reason?

@Jim The hope is to retain the ability to get feedback from vzn and other where it would be useful (like english grammar), by not cluttering their room and forcing them to ignore us.

agreed , you are probably at office now right? when will we resume ? I am eager, and I am going to disturb you , since I have no hope except you. I promise you, the third part will be quick, 10 to 15 min, but the second part is cruicial, so.. would you please find some time today?

@Jim I browse through it right now, but that doesn't mean that I will immediately understand it. Why is it important that I understand it today?

Nothing particular, I am being impatient! ok take you time and ask straight if anything is unclear, other person who will read the draft, will not be as patient as you, so my objective is to make it easy as much as I could.

Digression: did you forget to vote something? (being cryptic intentionally).

@Jim ??? I did vote up vzn's community promotion ad, but it has seven upvotes and four downvotes now. The additional downvotes probably happened, because it was shown as ad, people are encouraged to vote on the ad there, and they were annoyed that the ad didn't immediately tell them that it belonged to the CS chat room.

ok.

@Jim The statement "If $G_k \simeq H_k$ then ..." suggests that you believe this to be a likely event. However, the procedure by which you constructed $G_k$ and $H_k$ doesn't support your optimism to end up in a case where they are indeed isomorphic. And even if you were lucky and they are isomorphic, this doesn't mean that this local isomorphism is induced by any global isomorphism.

@ThomasKlimpel "However, the procedure by which you constructed $G_k$ and $H_k$ doesn't support your optimism to end up in a case where they are indeed isomorphic. "--- to be sure (so I can answer you properly), how I constructed $H_k$? is the construction of $H_k$ is geven in paragraph 1,2 and 4 on page 3 ?

@Jim If you don't use the same procedure for the construction of $H_k$ that you used for $G_k$, then everything is fine. Then you don't need to answer me (except by confirming that the procedure for the construction of $H_k$ is different from the one for $G_k$).

@ThomasKlimpel If you think the construction of $H_k$ is given in paragraph 1,2 and 4 on page 3 : the construction of $H_k$ is given in paragraph 1,2 and 4 on page 3, actually provides a "reference positions of vertices" from where we start applying our generated permutations. It is more like a "reference position" . We "tune " it by applying permutation obtained from SEARCH tree which, in some way, "follow" the construction of $G_k$ and finds all possible permutations...

.... assuming that $H \simeq G$.

@ThomasKlimpel "If you don't use the same procedure for the construction of $H_k$ that you used for $G_k$" --- initially no, but note that permutations are generated using the construction of $G_k$.

@ThomasKlimpel if it is confusing, for the time being, would you please Ignore first 2 lines of the 5th paragraph on page 3, and move on to remaining part, and tell me what is not clear in the remaining part?

@ThomasKlimpel I don't use the same procedure for the construction of $H_k$ that I used for $G_k$ but the construction of permutations for $H_k$ follows the similar procedure as $G_k$.

@ThomasKlimpel , are you still reading or done for the day?