Mathematics | Independent Sets, Covering and Matching

Mathematics | Independent Sets, Covering and Matching

1. Independent Sets

• A set of vertices I is called an independent set if no two vertices in set I are adjacent to each other in other words the set of non-adjacent vertices is called an independent set.
• It is also called a stable set.
• The parameter α₀(G) = max { |I|: I is an independent set in G } is called the independence number of G i.e. the maximum number of non-adjacent vertices.
• Any independent set I with |I| = α₀(G) is called a maximum independent set.

For above given graph G, Independent sets are:

I1 = {1}, I2 = {2}, I3 = {3}, I4 = {4}
I5 = {1, 3} and I6 = {2, 4} 

Therefore, the maximum number of non-adjacent vertices i.e Independence number α

₀

(G) = 2.

2. Vertex Covering

• A set of vertices K which can cover all the edges of graph G is called a vertex cover of G i.e. if every edge of G is covered by a vertex in set K.
• The parameter β₀(G) = min { |K|: K is a vertex cover of G } is called the vertex covering a number of G i.e. the minimum number of vertices that can cover all the edges.
• Any vertex cover K with |K| = β₀(G) is called a minimum vertex cover.

For above given graph G, Vertex cover is:

V1 = {1, 3}, V2 = {2, 4}, 
V3 = {1, 2, 3}, V4 = {1, 2, 3, 4}, etc.  

Therefore, minimum number of vertices which can cover all edges, i.e., Vertex covering number β

₀

(G) = 2.

Notes -

• I is an independent set in G if V(G) - I is vertex cover of G.
• For any graph G, α₀(G) + β₀(G) = n, where n is number of vertices in G.

Edge Covering -

• A set of edges F which can cover all the vertices of graph G is called a edge cover of G i.e. if every vertex in G is incident with a edge in F.
• The parameter β₁(G) = min { |F|: F is an edge cover of G } is called edge covering number of G i.e sum of minimum number of edges which can cover all the vertices and number of isolated vertices(if exist).
• Any edge cover F with |F| = β₁(G) is called a minimum edge cover.

For above given graph G, Edge cover is:

E1 = {a, b, c, d}, 
E2 = {a, d} and E3 = {b, c}.  

Therefore, minimum number of edges which can cover all vertices, i.e., Edge covering number β

₁

(G) = 2.

Note -

For any graph G, α

₁

(G) + β

₁

(G) = n, where n is number of vertices in G.

3. Matching -

• The set of non-adjacent edges is called matching i.e independent set of edges in G such that no two edges are adjacent in the set.
• he parameter α₁(G) = max { |M|: M is a matching in G } is called matching number of G i.e the maximum number of non-adjacent edges.
• Any matching M with |M| = α₁(G) is called a maximum matching.

For above given graph G, Matching are:

M1 = {a}, M2 = {b}, M3 = {c}, M4 = {d}
M5 = {a, d} and M6 = {b, c} 

Therefore, maximum number of non-adjacent edges i.e matching number α

₁

(G) = 2.

Complete Matching:

A matching of a graph G is complete if it contains all of G'svertices. Sometimes this is also called a perfect matching.

HALL’S MARRIAGE THEOREM:

The bipartite graph G =(V, E) with bipartition (V1, V2) has a complete matching from V1 to V2 if and only if |N (A)| > |A| for all subsets A of V1. (This is both necessary and sufficient condition for complete matching.)

Solved Examples

Example 1: Finding an Independent Set

Problem: Find an independent set in the following graph G:

lua
Copy code
    1---2
    |   |
    3---4

Solution: Independent sets: I1={1}, I2={2}, I3={3}, I4={4}, I5={1,3}, I6={2,4}.
Maximum independent set: I5={1,3}or I6={2,4}
Independence number: α0(G)=2

Example 2: Finding a Vertex Cover

Problem: Find a vertex cover in the same graph G:

lua
Copy code
    1---2
    |   |
    3---4

Solution: Vertex covers: V1={1,3}, V2={2,4}, V3={1,2,3}, V4={1,2,3,4}.
Minimum vertex cover: V1={1,3} or V2={2,4}
Vertex covering number: β0(G)=2

Related Articles:

• Mathematics | Independent Sets, Covering and Matching
• Matching (Graph Theory)

Practice Problems

• Problem 1: A graph G has vertices labeled {1,2,3,4,5}{1,2,3,4,5}. The edges are {(1,2),(1,3),(2,4),(3,5)}{(1,2),(1,3),(2,4),(3,5)}. Find the independence number α0(G)α0​(G).
• Problem 2: For the same graph G in Problem 1, determine the minimum vertex cover β0(G)β0​(G).
• Problem 3: For the same graph G in Problem 1, find the edge covering number β1(G)β1​(G).
• Problem 4: For the same graph G in Problem 1, determine the maximum matching α1(G)α1​(G).
• Problem 5: A graph G has vertices labeled {A,B,C,D,E,F}{A,B,C,D,E,F}. The edges are {(A,B),(B,C),(C,D),(D,E),(E,F),(F,A)}{(A,B),(B,C),(C,D),(D,E),(E,F),(F,A)}. Find the independent sets of GG.
• Problem 6: For the same graph G in Problem 5, determine the minimum vertex cover.
• Problem 7: For the same graph G in Problem 5, find the edge covering number.
• Problem 8: For the same graph G in Problem 5, determine the maximum matching.
• Problem 9: Consider a complete graph K5​ with vertices {1,2,3,4,5}{1,2,3,4,5}. Find the independence number and minimum vertex cover.
• Problem 10: Consider a bipartite graph K3,4K3,4​ with partition sets {A,B,C}{A,B,C} and {D,E,F,G}{D,E,F,G}. Determine the matching number and edge covering number.