1. **Problem Statement:** We need to state and prove the necessary and sufficient condition for two graphs to be isomorphic and illustrate with examples. 2. **Definition of Graph Isomorphism:** Two graphs $G = (V_G, E_G)$ and $H = (V_H, E_H)$ are isomorphic if there exists a bijection $f: V_G \to V_H$ such that any two vertices $u, v \in V_G$ are adjacent in $G$ if and only if $f(u)$ and $f(v)$ are adjacent in $H$. 3. **Necessary and Sufficient Condition:** The necessary and sufficient condition for $G$ and $H$ to be isomorphic is the existence of a bijection $f$ between their vertex sets that preserves adjacency: $$\forall u,v \in V_G, \{u,v\} \in E_G \iff \{f(u),f(v)\} \in E_H$$ 4. **Proof:** - **Necessity:** If $G$ and $H$ are isomorphic, by definition there exists such a bijection $f$ preserving adjacency. - **Sufficiency:** If such a bijection $f$ exists, then $G$ and $H$ have the same structure, so they are isomorphic. 5. **Example:** Consider $G$ with vertices $\{1,2,3\}$ and edges $\{\{1,2\}, \{2,3\}\}$, and $H$ with vertices $\{a,b,c\}$ and edges $\{\{a,b\}, \{b,c\}\}$. Define $f$ by $f(1)=a$, $f(2)=b$, $f(3)=c$. Since adjacency is preserved, $G$ and $H$ are isomorphic. This condition ensures that the graphs have the same connectivity pattern, which is the essence of graph isomorphism.