Question

Prove that for two graphs \(G\) and \(H\) that \(G\) is isomorphic to \(H\) if and only if \(\bar{G}\) is isomorphic to \(\bar{H}\).

Short answer

Graph \(G\) is isomorphic to \(H\) if and only if \(\bar{G}\) is isomorphic to \(\bar{H}\).

Step-by-step solution

Understand Isomorphism

Two graphs are isomorphic if there exists a one-to-one mapping between their vertex sets that preserves adjacency. This means that if two vertices are connected by an edge in one graph, their corresponding vertices must also be connected in the other graph.

Define the Complement of a Graph

The complement of a graph \(G\), denoted as \(\bar{G}\), contains the same set of vertices as \(G\). However, two vertices are adjacent in \(\bar{G}\) if and only if they are not adjacent in \(G\).

Propose the Isomorphic Mapping

Assume that \(G\) is isomorphic to \(H\). This means there is a bijection \(f: V(G) \to V(H)\) such that \((u, v)\) is an edge in \(G\) if and only if \((f(u), f(v))\) is an edge in \(H\).

Apply the Mapping to Complements

Using the same mapping \(f\), apply it to the complements. If \((u, v)\) is not an edge in \(G\), then \((f(u), f(v))\) is not an edge in \(H\). Thus, \(f\) also maps \(\bar{G}\) to \(\bar{H}\), proving \(\bar{G}\) is isomorphic to \(\bar{H}\).

Verify the Reverse Direction

Now, assume \(\bar{G}\) is isomorphic to \(\bar{H}\), meaning there is a bijection \(g: V(\bar{G}) \to V(\bar{H})\) preserving non-adjacency. This implies the same \(g\) preserves adjacency in the original graphs, hence \(G\) is isomorphic to \(H\).

Conclusion

Since we have shown both directions of the implication, we conclude that two graphs \(G\) and \(H\) are isomorphic if and only if their complements \(\bar{G}\) and \(\bar{H}\) are isomorphic.

Key concepts

Graph Complements

In graph theory, the concept of a graph complement is essential to understanding various properties related to isomorphism. A graph's complement, denoted as \( \bar{G} \), involves a transformation that affects the edge structure while keeping the vertex set intact. This means that for any two vertices in the original graph \( G \), they are interconnected by an edge in the complement \( \bar{G} \) if they were not connected in the original graph \( G \), and vice versa.

When dealing with problems that involve isomorphism between graph complements, it is crucial to grasp this change in edge relationships. The transformation flips the scenario of adjacency among vertices. This property is valuable because it allows investigation of isomorphic properties between graphs by simply peering into their complements. Demonstrating that the complements \( \bar{G} \) and \( \bar{H} \) are isomorphic can directly infer the same for \( G \) and \( H \). Thus, the concept of graph complements becomes a instrumental tool in proving equivalence and in revealing hidden symmetries between graphs.

Graph Theory

Graph theory is a rich and fascinating area of mathematics that deals with the study of graphs, which are abstract structures used to model pairwise relations between objects. A graph is composed of vertices (or nodes) and edges (or arcs) that connect pairs of vertices.

One of the central notions in graph theory is *isomorphism*. Graph isomorphism is a form of equivalence between graphs. Two graphs \( G \) and \( H \) are considered isomorphic if there exists a bijection between the vertex sets of \( G \) and \( H \) that preserves the connectivity, i.e., the adjacency of vertices. This equivalence makes isomorphic graphs indistinguishable in terms of their structural properties, even if they appear different at first glance.

Graphs have numerous applications across different fields such as computer science, biology, and social sciences, making graph theory an instrumental branch of discrete mathematics. Understanding the fundamentals of graph theory, like isomorphism, graph complements, and mappings, provides powerful insights into complex datasets and networks.

Bijective Mapping

In the context of graph isomorphism, bijective mapping is a pivotal concept. A bijection is a function between two sets that is both injective (one-to-one) and surjective (onto), meaning that every element in the first set is paired with a unique element in the second set, and every element of the second set has exactly one corresponding element in the first set.

When it comes to graphs, a bijective mapping between the vertex sets of two graphs \( G \) and \( H \) allows us to analyze if the graphs are isomorphic. This mapping must preserve the edge structure, which means if two vertices are connected by an edge in \( G \), the corresponding vertices in \( H \) should also be attached by an edge under the mapping.

Moreover, when examining the complements of graphs, the same bijection helps in demonstrating the equivalence. The bijection applied to \( \bar{G} \) and \( \bar{H} \) must maintain non-adjacency just as it preserves adjacency in the original graphs. This bijective connection is what bridges graph structures and enables the comparison of their equivalencies.