Graph minor theory

Graph minor theory is a significant area of research within graph theory that deals with the concept of "minors." A graph \( H \) is said to be a minor of a graph \( G \) if \( H \) can be formed from \( G \) by a series of operations that include: 1. **Deleting vertices**: Removing a vertex and its associated edges. 2. **Deleting edges**: Removing edges between vertices.

As of my last knowledge update in October 2023, "Apex Graph" does not refer to a widely recognized term or concept in mathematics, computer science, or any other specific field. It may refer to a specific software, tool, or framework that has emerged or gained popularity after that date, or it could also be a term used in a niche context, such as a specific application of graph theory, data visualization, or a particular software library.

Bounded expansion is a concept in graph theory that pertains to the behavior of certain classes of graphs, particularly in relation to their structure and properties. A family of graphs is said to have bounded expansion if, roughly speaking, the density of the graphs in the family does not grow too quickly as the size of the graphs increases.

In graph theory, a **bramble** is a concept used to describe a certain type of structure in a graph related to covering and dominating sets. Specifically, a bramble is a collection of subsets of vertices that captures the idea of a "tangled" set of vertices that cannot be separated from each other without removing some edges from the graph.

The **clique-sum** is a graph operation used primarily in graph theory, particularly in the study of graph properties and constructions. This operation allows you to combine two graphs in a way that preserves some of their characteristics while introducing new structure. Here’s how the clique-sum operates: 1. **Graphs Involved**: You start with two graphs, say \(G_1\) and \(G_2\).

The concept of a graph minor is a fundamental notion in graph theory, particularly in the study of graph structure and graph algorithms. A graph \( H \) is said to be a **minor** of another graph \( G \) if \( H \) can be formed from \( G \) by performing a series of operations that includes: 1. **Edge Deletion**: Removing edges from the graph. 2. **Vertex Deletion**: Removing vertices and incident edges from the graph.

Halin's Grid Theorem is a result in graph theory that describes the structure of certain infinite graphs. Specifically, it focuses on a type of infinite graph known as a "grid" graph, which is a graph that resembles a two-dimensional grid or lattice. Halin's theorem provides conditions under which such infinite grid graphs can be embedded into three-dimensional space without crossings.

A K-tree (or K-ary tree) is a type of tree data structure in which each node can have at most K children. This means that each node can link to K different nodes or child nodes, making it suitable for various applications where a more extensive branching factor is desirable compared to binary trees (which have a maximum of two children per node).

The Kelmans–Seymour conjecture is a conjecture in graph theory that relates to the structure of certain types of graphs. Specifically, it deals with the behavior of complete graphs and the existence of specific subgraphs within them. Formulated by Paul Kelmans and Neil Seymour, the conjecture states that every 2-edge-connected graph can be represented as a graph obtained from a complete graph by the contraction of edges.

A Partial k-tree is a data structure used primarily in the field of combinatorial optimization and computer science, particularly in topics related to the representation of combinatorial objects or configurations, such as combinations, subsets, or sequences. In general, a k-tree is a tree structure that represents all possible configurations of k elements chosen from a larger set, and it can be used for various applications, including generating combinations or permutations.

The term "Petersen family" can refer to different contexts depending on the specific area of interest. Here are a few possibilities: 1. **Cultural or Historical Context**: The Petersen family could refer to a family of historical or cultural significance in a specific region or country. 2. **Literary or Film Reference**: There might be fictional works, books, or movies that feature a "Petersen family" as characters.

A planar cover in the context of graph theory refers to a specific kind of covering or representation of a graph in a planar manner. Here are two common interpretations of "planar cover": 1. **Planar Graph**: If you have a graph that is planar, it means that it can be drawn on a plane without any edges crossing. A planar cover of a graph may refer to a way of embedding or representing the graph in the plane such that it maintains its properties without crossings.

"Shallow minor" is not a standard term widely recognized in music theory or other disciplines. However, the phrase could be interpreted in a few ways depending on the context in which it's used: 1. **Musical Context**: If we're discussing music, it might refer to a minor key (like A minor, B minor, etc.) that feels less intense or lacks depth, possibly due to its simplicity in composition or harmony.