Justify if the graphs shown below are isomorphic or not. [figure in the original paper] [figure in the original paper]

Since the original question does not provide the actual graphs, I will assume two common examples of graphs used in isomorphism tests for this problem. Let’s consider the following two graphs for justification:

Graph 1 (G₁)

      A
     / \
    B---C
     \ /
      D

Graph 2 (G₂)

      X
     / \
    Y---Z
     \ /
      W

---

Step-by-Step Justification

1. Definition of Isomorphism

Two graphs $G_1$ and $G_2$ are isomorphic if there exists a one-to-one correspondence (bijection) between their vertex sets such that:

• Two vertices are adjacent in $G_1$ if and only if their corresponding vertices are adjacent in $G_2$.
• The degree sequence of both graphs must be identical.

---

2. Degree Sequence Analysis

First, we determine the degree of each vertex in both graphs.

For Graph 1 (G₁):

• $A$: degree 2 (connected to B and C)
• $B$: degree 3 (connected to A, C, and D)
• $C$: degree 3 (connected to A, B, and D)
• $D$: degree 2 (connected to B and C)

Degree sequence of $G_1$: {2, 3, 3, 2}

For Graph 2 (G₂):

• $X$: degree 2 (connected to Y and Z)
• $Y$: degree 3 (connected to X, Z, and W)
• $Z$: degree 3 (connected to X, Y, and W)
• $W$: degree 2 (connected to Y and Z)

Degree sequence of $G_2$: {2, 3, 3, 2}

Since the degree sequences match, we proceed to check adjacency.

---

3. Vertex Correspondence and Adjacency Check

We attempt to map vertices of $G_1$ to $G_2$ such that adjacency is preserved.

Possible mapping:

• $A\leftrightarrow X$ (both have degree 2)
• $B\leftrightarrow Y$ (both have degree 3)
• $C\leftrightarrow Z$ (both have degree 3)
• $D\leftrightarrow W$ (both have degree 2)

Now, verify adjacency:

• In $G_1$, $A$ is adjacent to $B$ and $C$. In $G_2$, $X$ is adjacent to $Y$ and $Z$. ✅
• In $G_1$, $B$ is adjacent to $A$, $C$, and $D$. In $G_2$, $Y$ is adjacent to $X$, $Z$, and $W$. ✅
• Similarly, all other adjacencies match.

---

4. Conclusion

Since:

1. The degree sequences of both graphs are identical.
2. There exists a bijection between vertices that preserves adjacency.

The two graphs are isomorphic.

---

Visual Representation of Graphs and Their Isomorphism