Draw all connected graphs of order $5$ in which the distance between every two distinct vertices is odd.

Question

I'm working in the following graph theory excercise:

Draw all connected graphs of order $5$ in which the distance between every two distinct vertices is odd. Explain why you know that you have drawn all such graphs.

Is there another graph different to the complete graph of order $5$? Thanks in advance for any hint or help.

Answer 1

Hint: If $d(u_0,v)=n\gt0$ then $u_0$ has a neighbor $u_1$ such that $d(u_1,v)=n-1$.

Answer 2

Hint: for points abcde without loss of generality ab are connected. wlog bc are connected. The current distance between a and c is 2 this can only be made odd by connecting ac. Next step wlog cd are connected...