I am taking a course on Graph Theory in the upcoming semester so I started reading through Bondy and Murty.
Would rotating a graph that is an isomorphism of two graphs still count as another isomorphism, albeit not the most imaginative one? Or am I completely missing the point of this exercise by just wanting to take this lazy solution?
Edit:
Isomorphic graphs are graphs that are the same in structure, their edges and vertices, but differ in labels. The exercise itself says to find another isomorphism, so another way of redrawing the set of edges and vertices, between two graphs that are given.
I omitted the exercise because I thought my question wasn't dependent on it as I also didn't think providing the given graphs for the exercise was necessary either.
This is my first post on StackExchange so please be patient.
Thanks.
Best Answer
An isomorphism of graphs $G$ and $H$ is a function $f:V(G)\to V(H)$ that preserves adjacencies and non-adjacencies. Two graphs are said to be isomorphic if there exists an isomorphism.
Imagine $G$, a 4-cycle imbedded in the plane, as a square whose vertices are labeled $1,2,3,$ and $4$ clockwise from the upper left vertex. If we have another 4-cycle $H$ that is also imbedded as a square and whose vertices are labeled clockwise $1,2,3$ and $4$ starting with the upper right vertex, then the identity map serves as a natural isomorphism for these two graphs. Observe that $H$ is simply $G$ rotated by $90^{\circ}$.