# 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 iff 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.) Attention reader! Don’t stop learning now. Get hold of all the important CS Theory concepts for SDE interviews with the CS Theory Course at a student-friendly price and become industry ready.

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