I am studying graph theory from the book of Harary. There is a proof given in the book for the following:
If G is a connected graph and $p=q+1$ then G is acyclic, where, $p$= No. of points in the graph and $q$= No. of lines in the graph.
Proof given is as follows:
Assume that G has a cycle of length $n$. Then there are $n$ points and $n$ lines on the cycle and for each of the $p-n$ points not on the cycle, there is an incident line on a geodesic to a point of the cycle. Each such line is different, so, $q\ge p$, which is a contradiction.
I am facing difficulty in understanding the proof since it is too brief for me to understand. Can anyone explain me the line For each of the $p-n$ points not on the cycle, there is an incident line on a geodesic to a point of the cycle. Each such line is different.
I have just started with the basic definitions of graph and a few of its properties. It will be helpful if someone explains the above proof with an example pictorially.
Thankyou for the help!