Graph complement Ḡ: same vertices, complementary edge set (E(Ḡ) = E(K_n)\E(G)). |E(G)|+|E(Ḡ)| = C(n,2). Key duality: α(G)=ω(Ḡ), χ(G)·α(G)≥n. Self-complementary: G≅Ḡ (C_5, P_4 are self-complementary).
How to Use
Select graph:
- Ḡ: Complement graph
- Edges: C(n,2)-|E|
- Self: G≅Ḡ?
Complement Duality
α(G)=ω(Ḡ): max independent set in G = max clique in complement. χ(G)·α(G)≥n: fundamental bound. For perfect graphs: χ(G)=ω(G) and Ḡ is also perfect (WPGT). Complement preserves perfection!
Self-Complementary
G≅Ḡ requires n≡0,1(mod4) and |E|=n(n-1)/4. Examples: P_1, P_4, C_5. For n=4: unique (P_4). For n=5: two (C_5, bull). The number grows rapidly with n.
Step-by-Step Instructions
- 1Select graph.
- 2Compute Ḡ.
- 3Count edges.
- 4Check self-complementary.
- 5Compare α,ω.