## Independent set in discrete mathematicsThe independent set can be described as a set which contains the following details: - The independent set must not have any vertices which are adjacent to each other. The two vertices of the set must not have the same edge. The independent set can also be determined with the help of calculating a set of non-adjacent vertices in G.
- The independent set must not have any edges which are adjacent to each other. The two edges of the set must not have the same vertex.
- The independent set is also known as the stable set.
- The parameter α
_{0}(G) = max |I| can be known as the independence number of G. Here, I is used to show the independent set of G. The independent number can also be known as the maximum number of non-adjacent vertices. - If there is an independent set I with |I| = α
_{0}(G), then it will be known as the maximum independent set. The example of an independent set is described as follows:
From the above graph, we have the following details of an independent set. I1= {x} I2 = {y} I3 = {z} I4 = {u} I5 = {x, z} I6 = {y, u} Therefore, the maximum number of non-adjacent vertices or independence number α ## Independent line setSuppose there is a graph G = (V, E). If there is a subset L and no two edges in this subset are adjacent, in this case, the subset L of E will be known as the independent line set of G. The example of an independent line set is described as follows: From the above graph, we will consider the following subset. L1= {x, y} L2 = {x, y} {z, v} L3 = {x, u} {y, z} In the above graph, we can see that subsets L2 and L3 do not contain the adjacent edges. Hence the subsets L2 and L3 are the independent line sets. But the subset L1 is not an independent line set because we can form an independent line set with the help of minimum of two edges, but in L1, there is only one edge. Hence we can say that the subset L1 is not an example of an independent line set. ## Maximal Independent line setThe maximal independent line set is also known as the maximal matching. An independent line set will be known as the maximal independent line set of G if there is no possibility to add any extra edge of G into subset L. The example of a maximal independent line set is described as follows: From the above graph, we will consider the following subset. L1= {x, y} L2 = {{y, v} {z, f}} L3 = {{x, v} {y, z}, {u, f}} L4 = {{x, y} {z, f}} The subsets L2 and L3 are the maximal independent line sets because, in the above graph, we can see that subsets L2 and L3 are not able to add any other edge which is not adjacent. Hence we can say that the subsets L2 and L3 are known as the maximal independent line sets. ## Maximum independent line setWe can calculate the maximum independent line set of G with the help of calculating the maximum number of edges in that graph. We can calculate the maximum independent line set with the help of following formula: Number of edges in a maximum independent line set of G (β ) = matching number of G = Line independent number of G The example of a maximum independent line set is described as follows: From the above graph, we will get the following details about the subset. L1= {x, y} L2 = {{y, v} {z, f}} L3 = {{x, v} {y, z}, {u, f}} L4 = {{x, y} {z, f}} In the above graph, we can see that the subset L3 have the maximum edges and these edges are not adjacent. Hence the subset L3 can be called the maximum independent line set. The maximum independent line set of subset L3 can be represented like this: β1 = 3 ## Note: If there is a graph G which does not have an isolated vertex, then it will beα1 + β1 = number of vertices in a graph = |V|
Line independent number = β ## Edge Covering- An edge covering of a graph G can be described as a set of edges F, which is able to cover all the vertices of G. In the process of edge covering, every vertex of a graph G is incident with an edge in the set F.
- The parameter β
_{1}(G) = min |F| can be known as the edge covering number of a graph G. Here, F is used to show an edge cover of G. The edge covering number can also be known as the addition of a number of isolated vertices (if exists) and the minimum number of edges that is able to cover all the vertices. - If there is an edge cover F with |F| = β
_{1}(G), then it will be known as the minimum edge cover. The example of edge covering is described as follows:
From the above graph, we have the following details of edge covering. E1= {x, y, z, u} E2 = {x, u} E3 = {y, z} Therefore, the edge covering number or the minimum number of edges that covers all the vertices β ## Note: For a graph G, α |