I'm working in the following graph theory excercise:
Let $G_1$ and $G_2$ be two graphs with the same degree sequence If $G_1$ contains a vertex of degree $2$ that is adjacent to a vertex of degree $3$ and a vertex of degree $4$, while $G_2$ contains a vertex of degree $2$ that is adjacent to two vertices of degree $3$, can we conclude that $G_1 \not ≅ G_2$?
I've been trying to find a graph with those conditions but I can't certainly said why it doesn't exists, any hint or help will be really appreciated.
To make the question answered.
If we can then we’ll obtain a contradiction as follows. For each $i$ let $G_i’$ be any graph satisfying the problem conditions for the graph $G_i$ and $G$ be a union of disjoint copies of $G_i’$. Then $G$ satisfies the conditions both for $G_1$ and $G_2$, so we can put $G_1=G_2=G$.