Isomorphic Graphs
🔹 What Does “Isomorphic” Mean?
The word isomorphic comes from Greek:
- iso → same
- morph → shape or form
So, isomorphic graphs are graphs that have the same form.
This doesn’t mean the drawings must look identical.
It only means:
- They have the same number of vertices
- They have the same number of edges
- Their vertices are connected in the same pattern
The only difference is how they are drawn on paper.
🎯 A Simple Way to Think About It
Imagine you rearrange the furniture in your room.
The items are the same, and the way they relate (chair near table, bed beside window) is the same — only the positions changed.
Your room looks different, but it’s still the same room setup.
That’s exactly what isomorphic graphs are like.
🖼️ Simple Diagram to Visualize Isomorphism
Graph 1
A ----- B
| |
| |
D ----- CGraph 2
1
/ \
4 2
\ /
3At first glance, these two drawings look unrelated.
But let’s compare their structure:
- A connects to B and D
- B connects to A and C
- C connects to B and D
- D connects to A and C
And in Graph 2:
- 1 connects to 2 and 4
- 2 connects to 1 and 3
- 3 connects to 2 and 4
- 4 connects to 1 and 3
If we rename the vertices like this:
A → 1
B → 2
C → 3
D → 4
You will notice that every connection perfectly matches.
So these two graphs are isomorphic.
🌱 How Do You Check If Two Graphs Are Isomorphic?
This part confuses a lot of students, so let’s make it super simple.
To check isomorphism, we basically ask:
✔️ Do they have the same number of vertices?
If not — not isomorphic.
✔️ Do they have the same number of edges?
If not — not isomorphic.
✔️ Do the degrees of each vertex match?
(Degree = number of edges connected to a vertex)
For example, if Graph 1 has vertices with degrees:
2, 2, 2, 2
and Graph 2 has:
3, 2, 1, 1
→ They cannot be isomorphic.
✔️ Can you pair the vertices in a way that preserves the connections?
This is the heart of isomorphism.
Think of it like matching two sets of puzzle pieces.
If every vertex in Graph A can find a “partner” in Graph B with the same connections, then the graphs are isomorphic.
🎓 Real-Life Analogy
Think of two groups of friends.
In both groups:
- Person 1 is close to Person 2 and 3
- Person 2 is close to Person 1 and 4
- Person 3 is close to Person 1 and 4
- Person 4 is close to Person 2 and 3
Even if the people are different, the friendship pattern is identical.
So the “friendship network” of group A is isomorphic to group B.
The individuals differ → but the relationship structure matches.
🧠 Why Are Isomorphic Graphs Important?
Because in computer science, the shape of the graph matters more than the labels on it.
Isomorphism helps us:
- Detect equivalent network structures
- Compare chemical compound structures
- Check if two workflows do the same thing
- Reduce complex graphs by recognizing similarities
Understanding isomorphic graphs improves your ability to “see patterns” beyond surface appearance.
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