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# Complement of Graph

Let *'G−'* be a simple graph with some vertices as that of 'G' and an edge {U, V} is present in *'G−'*, if the edge is not present in G. It means, two vertices are adjacent in *'G−'* if the two vertices are not adjacent in G.

If the edges that exist in graph I are absent in another graph II, and if both graph I and graph II are combined together to form a complete graph, then graph I and graph II are called complements of each other.

## Example

In the following example, graph-I has two edges 'cd' and 'bd'. Its complement graph-II has four edges.

Note that the edges in graph-I are not present in graph-II and vice versa. Hence, the combination of both the graphs gives a complete graph of 'n' vertices.

**Note** − A combination of two complementary graphs gives a complete graph.

If 'G' is any simple graph, then

|E(G)| + |E(*'G-'*)| = |E(K_{n})|, where n = number of vertices in the graph.

## Example

Let 'G' be a simple graph with nine vertices and twelve edges, find the number of edges in *'G-'*.

You have, |E(G)| + |E(*'G-'*)| = |E(K_{n})|

12 + |E(*'G-'*)| =

9(9-1)2 = ^{9}C_{2}

12 + |E(*'G-'*)| = 36

|E(*'G-'*)| = 24

'G' is a simple graph with 40 edges and its complement *'G−'* has 38 edges. Find the number of vertices in the graph G or *'G−'*.

Let the number of vertices in the graph be 'n'.

We have, |E(G)| + |E(*'G-'*)| = |E(K_{n})|

40 + 38 = n(n-1) 2

156 = n(n-1)

13(12) = n(n-1)

n = 13

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