Graph Minors

A graph minor is a subgraph that can be obtained from a given graph by deleting vertices and edges. It preserves the connectivity and other structural properties of the original graph.

Graph minors are closed under taking minors, subgraphs, and edge contractions, but not under taking complements. The complement of a graph minor is not necessarily a graph minor.

Every graph has an infinite number of minors because there are infinitely many ways to delete vertices and edges from a graph. However, not all of these minors are distinct.

Graph minors preserve connectivity, meaning that if a graph is connected, then all of its minors are also connected. This is because deleting vertices and edges cannot create new connected components.

Depth-First Search (DFS) is commonly used to find graph minors. It involves systematically exploring the graph by traversing its edges in a depth-first manner. During the traversal, vertices and edges can be deleted to obtain minors.

Graph minors are significant in graph theory because they provide insights into the structure of graphs, are used to characterize graph classes, and are used to design efficient algorithms for graph problems.

Trees, planar graphs, and outerplanar graphs are all graph classes that are characterized by a forbidden minor. For example, trees are characterized by the fact that they do not contain the cycle graph $C_3$ as a minor.

Graph minors can be used to determine the chromatic number, maximum clique size, and minimum vertex cover of a graph. These parameters are all related to the structure of the graph, which can be analyzed using graph minors.

Graph minors have applications in network routing, circuit design, data compression, and other areas of computer science. They are used to model and analyze various network and computational structures.

The Hadwiger conjecture states that every graph with chromatic number $k$ contains a $k$-clique as a minor. It is one of the most famous unsolved problems in graph theory.

Every graph with treewidth $k$ has a minor that is a tree. This result is known as Robertson and Seymour's Graph Minors Theorem.

Graph minors can be used to determine whether a graph can be embedded in a surface, the genus of a graph, and the orientable genus of a graph. These parameters are related to the topological properties of graphs.

The Hadwiger conjecture is an open problem related to graph minors. It states that every graph with chromatic number $k$ contains a $k$-clique as a minor.

Graph minors can be used to design approximation algorithms and fixed-parameter tractable algorithms for NP-hard problems. These algorithms exploit the structural properties of graphs captured by graph minors.