Prove that: for a graph, G = (V, E) that I ⊆ V is the largest independent set if and only if V − I isthe smallest edge cover.

To prove this statement, we need to understand the definitions of independent set and edge cover in a graph.

1. Independent Set: An independent set in a graph is a set of vertices where no two vertices are adjacent. In other words, there is no edge between any two vertices in the set.

2. Edge Cover: An edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set.

Now, let's prove the statement:

(1) "I ⊆ V is the largest independent set if and only if V − I is the smallest edge cover."

We will prove this in two parts:

Part 1: If I is the largest independent set, then V - I is the smallest edge cover.

Proof: Suppose I is the largest independent set. Then, no vertex in I is connected to another vertex in I. Therefore, all edges in the graph must connect a vertex in I to a vertex in V - I. This means that V - I is an edge cover. Since I is the largest independent set, V - I must be the smallest set of vertices that can cover all edges. Therefore, V - I is the smallest edge cover.

Part 2: If V - I is the smallest edge cover, then I is the largest independent set.

Proof: Suppose V - I is the smallest edge cover. Then, every edge in the graph is incident to a vertex in V - I. Therefore, no edge can connect two vertices in I. This means that I is an independent set. Since V - I is the smallest edge cover, I must be the largest set of vertices that can be independent. Therefore, I is the largest independent set.

Therefore, we have proved that for a graph, G = (V, E), I ⊆ V is the largest independent set if and only if V − I is the smallest edge cover.