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 = ⁹C₂

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