The paragraph is given below:
My questions are:
1- Why the deformation retraction is a nonsymmetric notion while the homotopy equivalence is a symmetric one? symmetric and nonsymmetric with respect to what?
2-why we are sure that non of those 3 graphs is a deformation retract of any other?
Thanks!
A detail you probably missed is that "retract to something (by deformation or otherwise)" is a relation between a space $X$ and a subspace of it $A$. Therefore it is inherently asymmetric (because when $A\subseteq X$, $X\nsubseteq A$ unless $X=A$) and it doesn't hold between generic random spaces such as the three you mention. Those three spaces do not satisfy any mutual inclusion that I can see.