The problem asks you to prove that two graphs, $G_1$ and $G_2$, are isomorphic. Here’s the breakdown:

• Graph $G_1$ is formed by deleting an edge from $K_4$, the complete graph on 4 vertices.
• Graph $G_2$ is constructed by adding an edge to $K_{2,2}$, the complete bipartite graph with two sets of two vertices each, by joining two non-adjacent vertices.

Steps to prove that $G_1$ and $G_2$ are isomorphic:

1. Graph $G_1$:

• $K_4$ is the complete graph with 4 vertices, meaning every vertex is connected to every other vertex. It has $\binom{4}{2} = 6$ edges.
• Deleting one edge from $K_4$ gives a graph with 5 edges and 4 vertices. The degree sequence will be $(3, 3, 2, 2)$, meaning two vertices will have degree 3, and two will have degree 2.

2. Graph $G_2$:

• $K_{2,2}$ is a bipartite graph where vertices are split into two sets of size 2. Each vertex in one set is connected to both vertices in the other set, resulting in 4 edges total.
• Adding an edge between two non-adjacent vertices (i.e., vertices from the same set) gives a graph with 5 edges.
• The degree sequence of $K_{2,2}$ before adding an edge is $(2, 2, 2, 2)$. After adding the edge, the degrees of the two vertices connected by the new edge increase to 3, yielding a degree sequence of $(3, 3, 2, 2)$.

3. Isomorphism:

• Both graphs now have the same number of vertices (4) and edges (5).
• The degree sequence of both graphs is $(3, 3, 2, 2)$, meaning two vertices in each graph have degree 3, and two vertices have degree 2.
• Since both graphs have the same structure in terms of vertices, edges, and degree sequences, they are isomorphic.

Thus, $G_1$ and $G_2$ are isomorphic.

Would you like more details or have any further questions on this? Here are a few related questions for practice:

1. What is the significance of the degree sequence in graph isomorphism?
2. Can two graphs have the same degree sequence but not be isomorphic?
3. How would you construct an example where two graphs are not isomorphic despite similar structures?
4. What are other common methods to prove graph isomorphism apart from degree sequences?
5. How does the structure of bipartite graphs affect isomorphism proofs?

Tip: When comparing graphs for isomorphism, always check vertex counts, edge counts, and degree sequences as a first step!