iI so what properties G_i 's should have in common ?
That is H is a minor of G_1,H is also a minor of G_2 etc.
If H is a single vertex then it is the minor of every non-empty graph so as stated I don't think there are any conditions you can place on the G_i in general. If you want to place restrictions on H then you may be able to get some condition on the G_i's
That is correct i am looking who interested to work on the same problem
Also, every graph is a minor of itself. Thus, when H is a minor of ANOTHER graph G, there is no way the two can be isomorphic to each other.
That is exactly what i mean we need to find the non-isomorphic graphs G's as they they cannot be isomorphic this there any hereditary properties or not
Yes. And not only for the trivial reasons already given. For instance, if you make the problem harder, and ask, "Can a graph H be a minor obtained only by deleting a single edge of several non-isomorphic graphs?" The answer in yes, and 'several' can be specified to be greater than any chosen (finite) number: a path of n+2 vertices is the minor obtained by deleting a single edge of each graph formed by adding to it an edge from one of the leaves to each of the n non-adjacent vertices (as well as others obtained by adding an edge between two non-adjacent non-leaves).
Can a graph H be a minor obtained only by deleting a single edge of several non-isomorphic graphs?
That question is good we need to find some property existence from minor to graph.
So that if we can prove some property for a small graphs which could be minor we can extend it for the bigger graphs for which it is a minor under certain conditions.
Yes. For example, the path P3 is a minor of two non-isomorphic graphs P4 and C3.
"Can a graph H be a minor obtained only by deleting a single edge of several non-isomorphic graphs?"
Correcteducation version of question as suggested
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