Graph Theory - Complement Graphs

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Complement Graph

In graph theory, the complement graph is derived from a given graph that has the same set of vertices as the original graph, but the edges are different. In the complement graph, there is an edge between two vertices if and only if there is no edge between those two vertices in the original graph.

Specifically, the complement graph of a given graph −

G=(V,E) is a graph G =(V,E )

where,

• The vertex set V is the same in both graphs.
• The edge set E contains exactly the edges that are not present in the original graph E, but would exist if they were added to the graph.

In other words, if there is an edge between two vertices in the original graph G, then there is no edge between them in the complement graph G, and vice versa.

Properties of Complement Graphs

The complement graph has several properties that make it a useful concept in graph theory. These properties are −

• Vertex Correspondence: The vertex set of the complement graph is identical to the vertex set of the original graph.
• Edge Connection: There is an edge between two vertices in the complement graph if and only if there is no edge between them in the original graph.
• Number of Edges: If a graph G has m edges and n vertices, the complement graph G' will have n(n-1)/2 - m edges, since this is the total number of possible edges in a complete graph minus the edges in G.
• Complements of Complements: The complement of the complement of a graph G is the original graph G, i.e., (G')'' = G.

Constructing a Complement Graph

To construct the complement graph of a given graph G, follow these steps −

• Identify the vertex set V and edge set E of the original graph G:

  Let G = (V, E) be a graph with vertex set V = {v₁, v₂, ..., v_n} and edge set E consisting of pairs of vertices connected by edges.

• Create the complement graph G' by including all possible edges between the vertices that are not present in the original graph:

  In the complement graph G', for each pair of vertices (v_i, v_j) that do not have an edge in G, include the edge (v_i, v_j) in G'.

• Ensure that the complement graph contains no self-loops (edges between a vertex and itself). These are not included in either the original graph or the complement graph:

  Thus, for each vertex v_i, there will be no edge (v_i, v_i) in the complement graph.

Example

Consider the following graph G:

• Step 1: Identify the vertex set and edge set of G: V = {A, B, C, D}, E = {(A, B), (B, C), (C, D)}
• Step 2: Construct the complement graph G': The complement graph will have the same vertices, but the edges will be those that are not in E. In this case, G' will have edges (A, C), (A, D), (B, D).

Thus, the complement graph G' would look like this:

In this example, edges are added between vertices that do not share an edge in the original graph G.

Applications of Complement Graphs

Complement graphs have various applications in graph theory and computer science, they are −

• Finding Independent Sets: In the complement graph, a clique corresponds to an independent set in the original graph. This can be useful for problems involving finding independent sets.
• Coloring Problems: The complement graph can be used to study the coloring properties of the original graph, especially in algorithms involving graph coloring.
• Network Design: Complement graphs can be used in network design problems where the complement graph represents alternative pathways or connections in a network.
• Graph Isomorphism: The complement graph can help in solving graph isomorphism problems by comparing the structure of the complement graphs.

In graph theory, a clique is a subset of vertices in a graph such that every two vertices in the subset are connected by an edge. In other words, a clique is a complete subgraph, where every possible edge exists between the vertices in the subset.

Special Types of Complement Graphs

Some special types of complement graphs arise from particular properties of the original graph. These are −

• Complement of a Complete Graph: The complement of a complete graph is an empty graph, as all vertices are already connected by edges.
• Complement of a Cycle Graph: The complement of a cycle graph forms a disconnected graph with multiple components.
• Complement of a Bipartite Graph: The complement of a bipartite graph is a complete bipartite graph, where each vertex in one set is connected to all vertices in the other set.

Complement of a Complete Graph

The complement of a complete graph K_n is an empty graph because every pair of vertices in a complete graph is connected by an edge. Therefore, the complement graph has no edges.

If the original graph G is a complete graph with n vertices, the complement graph G' will have no edges, as all pairs of vertices are already connected in G.

Complement of a Cycle Graph

The complement of a cycle graph C_n is a disconnected graph with multiple components. In a cycle graph, each vertex is connected to two other vertices, but the complement graph will connect vertices that are not adjacent in the cycle graph.

For example, the complement of a cycle graph with four vertices C₄ will consist of two disconnected edges.

Complement of a Bipartite Graph

The complement of a bipartite graph G = (V₁, V₂, E) is a complete bipartite graph. In a bipartite graph, vertices are divided into two disjoint sets, and edges connect vertices in different sets. The complement graph will have all possible edges between the two sets, forming a complete bipartite graph.