Below is a clear, simple, and exam-focused explanation of Isomorphism and Homomorphism in Graph Theory and Algebra, suitable for BCA/MCA/Engineering/Discrete Mathematics.
⭐ ISOMORPHISM AND HOMOMORPHISM
These two concepts describe relationships between mathematical structures (graphs, groups, semigroups, rings, etc.).
They tell us whether two structures are identical or structurally related.
⭐ PART 1: ISOMORPHISM
⭐ 1. What is an Isomorphism?
An isomorphism is a structure-preserving one-to-one correspondence between two algebraic structures or graphs.
In simple words:
Two structures are isomorphic if they look the same, even if drawn or named differently.
⭐ A. Graph Isomorphism (Graph Theory)
Two graphs (G_1 = (V_1, E_1)) and (G_2 = (V_2, E_2)) are isomorphic if:
There exists a bijection
[
f : V_1 \to V_2
]
such that:
[
(u, v) \in E_1 \iff (f(u), f(v)) \in E_2
]
✔ Meaning:
- Vertices correspond one-to-one
- Edges correspond one-to-one
- Shape/structure is identical
- Only labels or drawing position differ
⭐ Conditions to test graph isomorphism
Check the following:
- Same number of vertices
- Same number of edges
- Same degree sequence
- Same number of cycles
- Same pattern of connections
If all match → graphs are likely isomorphic.
⭐ Example (Graph Isomorphism)
Graph G1:
A—B—C
| |
D─────
Graph G2:
1—2—3
| |
4─────
These two are isomorphic (same square structure).
⭐ When graphs are not isomorphic?
- Different number of vertices
- Different number of edges
- Different degree sequences
- One has cycles of length 3, other does not
- One is connected, other disconnected
⭐ B. Isomorphism in Groups, Semigroups, Rings
Two groups (G) and (H) are isomorphic if there exists a bijective homomorphism:
[
f : G \to H
]
such that:
[
f(a * b) = f(a) * f(b)
]
✔ Meaning:
- Same algebraic structure
- Same number of elements
- Same operation behavior
- Elements may have different names
Notation:
[
G \cong H
]
⭐ PART 2: HOMOMORPHISM
⭐ 2. What is a Homomorphism?
A homomorphism is a structure-preserving map between algebraic structures.
It need NOT be one-to-one or onto.
In simple words:
A homomorphism preserves operation but may “compress” or “merge” elements.
⭐ A. Graph Homomorphism
A function
[
f : V_1 \to V_2
]
between graphs (G_1) and (G_2) is a graph homomorphism if:
[
(u, v) \in E_1 \Rightarrow (f(u), f(v)) \in E_2
]
✔ Meaning:
Edges map to edges, but not necessarily bijectively.
Used in:
- Graph coloring
- Constraint satisfaction
- Pattern matching
⭐ B. Group Homomorphism
A function
[
f : G \to H
]
is a group homomorphism if:
[
f(a * b) = f(a) * f(b)
]
Properties:
- Preserves identity
- Preserves inverse
- Kernel = elements mapped to identity
- Image is a subgroup of (H)
⭐ Kernel of a homomorphism
[
\ker(f) = {a \in G : f(a) = e_H}
]
Kernel helps build quotient groups.
⭐ Isomorphism = special homomorphism
Isomorphism = homomorphism + bijection.
⭐ Differences Between Isomorphism and Homomorphism
| Feature | Homomorphism | Isomorphism |
|---|---|---|
| Structure preservation | Yes | Yes |
| One-to-one? | Not required | Required |
| Onto? | Not required | Required |
| Reversible? | No | Yes |
| Purpose | Relate structures | Show structures are identical |
| Kernel | May exist | Always trivial (only identity) |
⭐ Applications
✔ Graph Isomorphism
- Pattern matching
- Computer vision
- Chemical structure analysis
- Cryptography
- Network comparison
✔ Homomorphism (Algebra)
- Building quotient structures
- Cryptography (RSA uses ring homomorphism idea)
- Automata theory
- Simplification of algebraic systems
✔ Graph Homomorphism
- Graph coloring
- Constraint satisfaction
- Checking substructures
⭐ Quick Exam-Oriented Summary
Isomorphism
- One-to-one, onto map
- Preserves structure
- Reversible
- Graphs: same structure, different drawing
- Groups: (f(a * b) = f(a) * f(b)), bijective
Homomorphism
- Operation-preserving map
- Not necessarily one-to-one
- Kernel exists
- Used to build quotient structures
- Graph homomorphism preserves edges