Let R be the reduction from 3-SAT to VERTEX-COVER as discussed in class. **Graph Explanation:**The graph shown is a bipartite graph consisting of two sets of nodes, connected by edges. The nodes are labeled as $ P, \neg P, Q, \neg Q, S, \neg S, T, \neg T $. The edges between the nodes indicate the relationships that need to be considered to construct the 3-CNF (Conjunctive Normal Form) formula.### Nodes:- **Positive Nodes:** $ P, Q, S, T $- **Negative Nodes:** $ \neg P, \neg Q, \neg S, \neg T $### Edges:The graph connects each variable and its negation:- $ P $ is connected to $ \neg P $- $ Q $ is connected to $ \neg Q $- $ S $ is connected to $ \neg S $- $ T $ is connected to $ \neg T $Additionally, other connections exist indicating clause construction for the formula:- Edges between positive and negative nodes suggest combinations in clauses.**Question:**(1) Show the 3-CNF formula for which R builds the following graph.**Answer Choices:**- $ \circ \ (P \land Q \land T) \lor (P \land Q \land S) \lor (T \land Q \land S) $- $ \circ \ (P \lor Q \lor T) \land (P \lor Q \lor S) \land (T \lor Q \lor S) $- $ \circ \ (\neg P \land Q \land T) \lor (\neg P \land Q \land S) \lor (T \land \neg Q \land S) $- $ \circ \ (\neg P \lor Q \lor T) \land (\neg P \lor Q \lor S) \land (T \lor \neg Q \lor S) $

In 3 CNF SAT we have at least 3 clauses and in the clauses we will have the almost 3 literals.

(1) Show the 3-CNF formula for which R builds the following graph. P -P T S -P 0 (P^Q ^T) V (P^Q ^S) V (T ^Q ^S) O (P V QVT) A (PVQVS) A (TV Q V S) O (PAQAT) V (P AQ AS) V (TA QAS) O (PV QVT) A (PV QV S) A (TV QV S) -S S T -T S

(1) Show the 3-CNF formula for which R builds the following graph. P -P T S -P 0 (P^Q ^T) V (P^Q ^S) V (T ^Q ^S) O (P V QVT) A (PVQVS) A (TV Q V S) O (PAQAT) V (P AQ AS) V (TA QAS) O (PV QVT) A (PV QV S) A (TV QV S) -S S T -T S

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Let R be the reduction from 3-SAT to VERTEX-COVER as discussed in class.

**Graph Explanation:**

The graph shown is a bipartite graph consisting of two sets of nodes, connected by edges. The nodes are labeled as $ P, \neg P, Q, \neg Q, S, \neg S, T, \neg T $. The edges between the nodes indicate the relationships that need to be considered to construct the 3-CNF (Conjunctive Normal Form) formula.

### Nodes:

- **Positive Nodes:** $ P, Q, S, T $
- **Negative Nodes:** $ \neg P, \neg Q, \neg S, \neg T $

### Edges:

The graph connects each variable and its negation:

- $ P $ is connected to $ \neg P $
- $ Q $ is connected to $ \neg Q $
- $ S $ is connected to $ \neg S $
- $ T $ is connected to $ \neg T $

Additionally, other connections exist indicating clause construction for the formula:

- Edges between positive and negative nodes suggest combinations in clauses.

**Question:**

(1) Show the 3-CNF formula for which R builds the following graph.

**Answer Choices:**

- $ \circ \ (P \land Q \land T) \lor (P \land Q \land S) \lor (T \land Q \land S) $
- $ \circ \ (P \lor Q \lor T) \land (P \lor Q \lor S) \land (T \lor Q \lor S) $
- $ \circ \ (\neg P \land Q \land T) \lor (\neg P \land Q \land S) \lor (T \land \neg Q \land S) $
- $ \circ \ (\neg P \lor Q \lor T) \land (\neg P \lor Q \lor S) \land (T \lor \neg Q \lor S) $

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