Complement of Graph

The article is already well-structured. I'll replace the JPG image with an SVG diagram showing the graph and its complement side by side.

The complement of a graph G, denoted as G̅, is a simple graph with the same set of vertices as G. An edge {U, V} exists in G̅ if and only if that edge is not present in G. In other words, two vertices are adjacent in G̅ if and only if they are not adjacent in G.

If the edges that exist in graph I are absent in graph II, and combining both graphs produces a complete graph, then graph I and graph II are called complements of each other.

Example: Finding the Complement Graph

In the following example, graph-I has two edges 'bd' and 'cd'. Its complement graph-II has the four remaining edges needed to form the complete graph K₄ −

The edges in graph-I are not present in graph-II and vice versa. Combining both graphs gives the complete graph on n vertices.

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

Formula for Complement Edges

If G is any simple graph with n vertices, the following relationship holds −

|E(G)| + |E(G̅)| = |E(K_n)| = n(n−1)/2

This is because the complete graph K_n contains every possible edge between n vertices, and every edge must belong to either G or its complement G̅, but not both.

Example: Finding Number of Edges in the Complement

Let G be a simple graph with 9 vertices and 12 edges. Find the number of edges in G̅.

Using the formula −

|E(G)| + |E(G̅)| = n(n-1)/2

12 + |E(G̅)| = 9(9-1)/2

12 + |E(G̅)| = 36

|E(G̅)| = 36 - 12

|E(G̅)| = 24

Example: Finding Number of Vertices

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

Let the number of vertices be n. Using the formula −

|E(G)| + |E(G̅)| = n(n-1)/2

40 + 38 = n(n-1)/2

78 = n(n-1)/2

156 = n(n-1)

156 = 13 × 12

n = 13

The graph G has 13 vertices.

Conclusion

The complement of a graph G contains exactly those edges that are missing from G. Together, G and G̅ always form a complete graph. The formula |E(G)| + |E(G̅)| = n(n−1)/2 allows you to find missing edge counts or vertex counts when the other values are known.