Given an undirected graph defined by the number of vertex **V** and the edges **E[ ]**, the task is to find **Maximal Independent Vertex Set** in an undirected graph.

Independent Set:An independent set in a graph is a set of vertices which are not directly connected to each other.

**Note:** It is a given that there is at least one way to traverse from any vertex in the graph to another, i.e. the graph has one connected component.

**Examples:**

Input:V = 3, E = { (1, 2), (2, 3) }Output:{1, 3}Explanation:

Since there are no edges between 1 and 3, and we cannot add 2 to this since it is a neighbour of 1, this is the Maximal Independent Set.

Input:V = 8,

E = { (1, 2), (1, 3), (2, 4), (5, 6), (6, 7), (4, 8) }Output:{2, 3, 5, 7, 8}

**Approach:**

This problem is an NP-Hard problem, which can only be solved in exponential time(as of right now).

Follow the steps below to solve the problem:

- Iterate through the vertices of the graph and use backtracking to check if a vertex can be included in the
**Maximal Independent Set**or not. - Two possibilities arise for each vertex, whether it can be included or not in the maximal independent set.
- Initially start, considering all vertices and edges. One by one, select a vertex. Remove that vertex from the graph,
*excluding it from the maximal independent set*, and recursively traverse the remaining graph to find the maximal independent set. - Otherwise, consider the selected vertex in the maximal independent set and remove all its neighbors from it. Proceed to find the maximal independent set possible excluding its neighbors.
- Repeat this process for all vertices and print the maximal independent set obtained.

Below is the implementation of the above approach:

## Python3

`# Python Program to implement` `# the above approach` ` ` `# Recursive Function to find the` `# Maximal Independent Vertex Set ` `def` `graphSets(graph):` ` ` ` ` `# Base Case - Given Graph ` ` ` `# has no nodes` ` ` `if` `(` `len` `(graph) ` `=` `=` `0` `):` ` ` `return` `[]` ` ` ` ` `# Base Case - Given Graph` ` ` `# has 1 node` ` ` `if` `(` `len` `(graph) ` `=` `=` `1` `):` ` ` `return` `[` `list` `(graph.keys())[` `0` `]]` ` ` ` ` `# Select a vertex from the graph` ` ` `vCurrent ` `=` `list` `(graph.keys())[` `0` `]` ` ` ` ` `# Case 1 - Proceed removing` ` ` `# the selected vertex` ` ` `# from the Maximal Set` ` ` `graph2 ` `=` `dict` `(graph)` ` ` ` ` `# Delete current vertex ` ` ` `# from the Graph` ` ` `del` `graph2[vCurrent]` ` ` ` ` `# Recursive call - Gets ` ` ` `# Maximal Set,` ` ` `# assuming current Vertex ` ` ` `# not selected` ` ` `res1 ` `=` `graphSets(graph2)` ` ` ` ` `# Case 2 - Proceed considering` ` ` `# the selected vertex as part` ` ` `# of the Maximal Set` ` ` ` ` `# Loop through its neighbours` ` ` `for` `v ` `in` `graph[vCurrent]:` ` ` ` ` `# Delete neighbor from ` ` ` `# the current subgraph` ` ` `if` `(v ` `in` `graph2):` ` ` `del` `graph2[v]` ` ` ` ` `# This result set contains VFirst,` ` ` `# and the result of recursive` ` ` `# call assuming neighbors of vFirst` ` ` `# are not selected` ` ` `res2 ` `=` `[vCurrent] ` `+` `graphSets(graph2)` ` ` ` ` `# Our final result is the one ` ` ` `# which is bigger, return it` ` ` `if` `(` `len` `(res1) > ` `len` `(res2)):` ` ` `return` `res1` ` ` `return` `res2` ` ` `# Driver Code` `V ` `=` `8` ` ` `# Defines edges` `E ` `=` `[ (` `1` `, ` `2` `),` ` ` `(` `1` `, ` `3` `),` ` ` `(` `2` `, ` `4` `),` ` ` `(` `5` `, ` `6` `),` ` ` `(` `6` `, ` `7` `),` ` ` `(` `4` `, ` `8` `)]` ` ` `graph ` `=` `dict` `([])` ` ` `# Constructs Graph as a dictionary ` `# of the following format-` ` ` `# graph[VertexNumber V] ` `# = list[Neighbors of Vertex V]` `for` `i ` `in` `range` `(` `len` `(E)):` ` ` `v1, v2 ` `=` `E[i]` ` ` ` ` `if` `(v1 ` `not` `in` `graph):` ` ` `graph[v1] ` `=` `[]` ` ` `if` `(v2 ` `not` `in` `graph):` ` ` `graph[v2] ` `=` `[]` ` ` ` ` `graph[v1].append(v2)` ` ` `graph[v2].append(v1)` ` ` `# Recursive call considering ` `# all vertices in the maximum ` `# independent set` `maximalIndependentSet ` `=` `graphSets(graph)` ` ` `# Prints the Result ` `for` `i ` `in` `maximalIndependentSet:` ` ` `print` `(i, end ` `=` `" "` `)` |

**Output:**

2 3 8 5 7

**Time Complexity:** O(2^{N})**Auxiliary Space:** O(N)

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