Independent Vertex Set

An independent vertex set of a graph G is a subset of vertices where no two vertices are adjacent (connected by an edge). This concept is the vertex counterpart of the independent line set (matching), and is fundamental to problems like graph coloring and vertex cover.

Independent Vertex Set

Let G = (V, E) be a graph. A subset S of V is called an independent vertex set of G if no two vertices in S are adjacent − that is, no edge in G connects any pair of vertices in S.

Example

Consider the following subsets −

S? = {e}               ? Only 1 vertex (trivially independent)
S? = {e, f}            ? Independent (no edge between e and f) ?
S? = {a, g, c}         ? Independent (no edges among a, g, c) ?
S? = {e, d}            ? Independent (no edge between e and d) ?

S₂, S₃, and S₄ are independent vertex sets because no two vertices in any of these subsets are adjacent.

Maximal Independent Vertex Set

An independent vertex set S is maximal if no other vertex of G can be added to S without creating an adjacency. Every vertex not in S must be adjacent to at least one vertex in S.

Example

Using the same graph −

S? = {e}               ? Not maximal (can add f or d)
S? = {e, f}            ? Maximal (cannot add any more vertex) ?
S? = {a, g, c}         ? Maximal (cannot add any more vertex) ?
S? = {e, d}            ? Not maximal (can add g)

S₂ and S₃ are maximal independent vertex sets. In S₁ and S₄, additional vertices can still be added without creating adjacency, so they are not maximal.

Maximum Independent Vertex Set

A maximal independent vertex set with the largest number of vertices is called the maximum independent vertex set. The number of vertices in it is called the vertex independence number, denoted by β₂.

Example

S? = {e, f}        ? 2 vertices
S? = {a, g, c}     ? 3 vertices  ? Maximum

?? = 3

S₃ = {a, g, c} is the maximum independent vertex set with β₂ = 3.

Relationship with Vertex Cover

For any graph G = (V, E), the vertex covering number (α₂) and vertex independence number (β₂) satisfy −

α₂ + β₂ = |V|

Also, if S is an independent vertex set of G, then (V − S) is a vertex cover of G.

Example: Complete Graph K_n

In K?, each vertex is adjacent to all other (n-1) vertices.

Maximum independent set can only contain 1 vertex:
    ?? = 1

Vertex covering number:
    ?? = |V| - ?? = n - 1

Verification: ?? + ?? = (n-1) + 1 = n = |V|  ?

Conclusion

An independent vertex set contains non-adjacent vertices. A maximal set cannot accept more vertices, and a maximum set has the largest possible size (β₂). The relationship α₂ + β₂ = |V| connects vertex independence with vertex covering for any graph.