独立集与顶点覆盖相互规约问题

Posted 2022-01-09 mb61caa1c74a413

目录

• 1.1 Theorem. INDEPENDENT-SET ≡ P VERTEX-COVER.
  • 1.1.1 证明充分性
  • 1.1.2 证明必要性

1.1 Theorem. INDEPENDENT-SET ≡ P VERTEX-COVER.

Pf. We show S is an independent set of size k iff V − S is a vertex cover
of size n – k.

如果图S有一个顶点个数为K的独立集存在的条件是：当且仅当V-S顶点覆盖的大小为n-k;

1.1.1 证明充分性

⇒

• Let S be any independent set of size k.
• V − S is of size n – k.
• Consider an arbitrary edge (u, v) ∈ E.
• S independent ⇒ either u ∉ S, or v ∉ S, or both.
  ⇒ either u ∈ V − S, or v ∈ V − S, or both.
  Thus, V − S covers (u, v).

1.1.2 证明必要性

⇐

• Let V − S be any vertex cover of size n – k.
• S is of size k.
• Consider an arbitrary edge (u, v) ∈ E.
• V − S is a vertex cover ⇒ either u ∈ V − S, or v ∈ V − S, or both.
  ⇒ either u ∉ S, or v ∉ S, or both.
  Thus, S is an independent set.