Mathematics | Independent Sets, Covering and Matching

1. Independent Sets –

• A set of vertices I is called independent set if no two vertices in set I are adjacent to each other or in other words the set of non-adjacent vertices is called independent set.
• It is also called a stable set.
• The parameter α₀(G) = max { |I|: I is an independent set in G } is called 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, maximum number of non-adjacent vertices i.e Independence number α₀(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 vertex covering number of G i.e the minimum number of vertices which 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 β₀(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 β₁(G) = 2.

Note – For any graph G, α₁(G) + β₁(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 α₁(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.)