Minimal property of a graph.

Question

A graph $G$ is said to be minimal if it loses property $P$ after deletion of an arbitrary edge.

I am considering a graph $G$ with edges $e_1,e_2,\ldots,e_n$ with property $P$. It loses its property only after deletion of edges $e_2$ and $e_n$. However, it retains property $P$ if we delete other edges $e_1,e_3,\ldots,e_{n-1}$. Can I say that graph $G$ is minimal? I am quite confused here. Kindly help. Thanks a lot for the help.

Answer 1

You are defining minimal as follows:

A graph G is said to be minimal if it loses property P after deletion of an arbitrary edge.

Now since in the graph you consider the property is retained after deleting of $e_1$, so your graph is not minimal. It would have been minimal if you couldn't find an edge whose removal retains the property.