While checking Isomorphism between two graphs, is it enough if I just check the degrees of all vertices and if the degrees of all vertices connected to every given vertex are identical in both graphs?
Is it also necessary to check for identical cycles in both graphs?
For example: https://i.stack.imgur.com/ptXaF.jpg
In the above image, it is possible to have a mapping between vertices of both graphs as both graphs have 2 vertices of degree 2 and 4 vertices of degree 3. Furthermore, every vertex of degree 3 is connected to 2 vertices of degree 3 and one vertex of degree 2 in both Graphs. Finally, the vertices with degree 2 are both connected to two vertices of degree 3.
But Graph 1 does not have a 3-vertex cycle like Graph 2. So are they isomorphic?