A degree-constrained spanning tree (DCST) is a specific type of spanning tree in a graph with the additional restriction that the degree (i.e., the number of edges connected) of each vertex must not exceed a specified limit. In other words, a DCST is a tree that spans all the vertices of a graph while ensuring that no vertex has a degree greater than a predefined upper bound.

The Kinetic Minimum Spanning Tree (KMST) is a concept derived from dynamic graph algorithms, specifically focusing on the minimum spanning tree (MST) in scenarios where the graph changes over time. In a typical minimum spanning tree problem, you have a weighted, undirected graph, and the goal is to find a tree that spans all vertices while minimizing the total edge weight. When the edges or weights of a graph change dynamically, maintaining an efficient representation of the minimum spanning tree becomes challenging.

A Minimum Degree Spanning Tree (MDST) is a variation of the Minimum Spanning Tree (MST) problem, which is typically concerned with connecting all vertices in a graph with the minimum possible total edge weight. In the context of an MDST, the objective shifts slightly. In an MDST, the goal is to find a spanning tree that not only minimizes the total edge weight but also limits the maximum degree of any vertex in the tree.

Multiple Spanning Tree Protocol (MSTP) is a network protocol used in Ethernet networks to prevent loops in network topologies while allowing for the efficient redundancy and load balancing of the network. Specifically, MSTP is an extension of the Spanning Tree Protocol (STP) and Multiple Spanning Tree Protocol (MSTP) to work across multiple VLANs (Virtual Local Area Networks).

A Rectilinear Minimum Spanning Tree (RMST) is a specific type of minimum spanning tree that is defined in a rectilinear (or grid-like) space, where the coordinates are aligned with the axes of a Cartesian plane. In a rectilinear geometry, the distance between two points is measured using the Manhattan distance (also known as the L1 distance), which is calculated as the sum of the absolute differences of their Cartesian coordinates.

A Trémaux tree, named after the French mathematician Édouard Trémaux, is a structure used in graph theory, specifically in the context of exploring undirected graphs. It is used to represent the exploration of the graph and the paths taken during a traversal. Typically, a Trémaux tree is constructed during a depth-first search (DFS) or a breadth-first search (BFS) of a graph, where the edges represent the paths followed by the traversal.

Sphere packing is a mathematical concept that involves arranging spheres in a way that maximizes the amount of space filled by the spheres without any overlapping. In a three-dimensional space, the goal is to determine how many identical spheres can be packed into a larger sphere (or, sometimes, just in space) in the most efficient manner.

Tripod packing, also known as tripod positioning, is a technique used primarily in the context of managing respiratory distress. It involves a person leaning forward while supporting themselves on their arms, typically positioned on their knees or in a standing position. This stance allows the individual to open up their chest and diaphragm, facilitating easier breathing. This position is often seen in patients experiencing severe asthma attacks, chronic obstructive pulmonary disease (COPD) exacerbations, or other conditions that compromise respiratory function.

Carathéodory's theorem is a fundamental result in convex geometry that characterizes the representation of points in a convex set.

The term "Tornado family" can refer to a few different contexts, but it most commonly pertains to either: 1. **Meteorology**: In meteorological terms, a "Tornado family" often describes a series of tornadoes that occur within the same storm system or weather event. Tornadoes can sometimes form in succession or in the same geographic area during a severe weather outbreak, and these may be referred to as part of a "family" because they share similar characteristics and conditions.

An Apollonian network is a type of geometric network that is constructed using a recursive process based on the properties of triangular tiling. It begins with a single triangle, which is then subdivided into smaller triangles recursively. The network has a rich structure and exhibits fractal characteristics, making it interesting in the study of complex networks.

Kinetic triangulation is a concept from computational geometry that deals with the dynamic problem of maintaining the properties of a triangulation of a set of points in motion. Specifically, it refers to the process of efficiently updating the triangulation structure as the points in the plane change their positions over time.

The Gap Theorem is a concept in the field of mathematics, particularly in the study of algebraic geometry and topology, though there are applications and related ideas in other areas of mathematics as well. In one of its forms, the Gap Theorem refers to a result concerning the existence of "gaps" in the spectrum of certain types of operators, particularly in the context of spectral theory.

A triangle mesh is a type of geometric representation commonly used in computer graphics, 3D modeling, and computational geometry. It consists of a collection of triangular faces that define a 3D shape or surface. Each triangle is typically defined by three vertices, which are points in 3D space, and the edges connecting these vertices.

The Even Circuit Theorem, often referred to in the context of graph theory and circuit design, primarily deals with the properties of circuits within graphs. While the term itself may not be universally defined across all disciplines, it is likely related to concepts in electrical engineering and theoretical computer science, where circuits can be represented as graphs. In general terms, in a graph: - A circuit (or cycle) is a closed path where no edges are repeated.

The Geiringer–Laman theorem is a result in the field of graph theory and combinatorial geometry, specifically concerning the rigidity of frameworks. The theorem provides a criterion for determining when a certain kind of graph, known as a "framework", can be considered rigid, meaning that its vertices cannot be moved without distorting the distances between them.

Kotzig's theorem is a result in graph theory concerning the properties of certain types of graphs, particularly related to edge colorings. Specifically, it states that every connected graph with a minimum degree of at least 3 can be decomposed into two spanning trees. This result is significant because spanning trees are foundational structures in graph theory, and their decomposition has implications for network design and reliability.

The Planar Separator Theorem is a concept in computational geometry and graph theory which states that for any planar graph, it is possible to partition the vertices of the graph into three disjoint sets: X, Y, and S. The sets have the following properties: 1. **Small Separator Size**: The size of the set S (the separator) is proportional to the square root of the number of vertices in the graph.

The Strong Perfect Graph Theorem, proved by Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas in 2006, establishes an important characterization of perfect graphs. The theorem states that a graph is perfect if and only if it contains no induced subgraph that is an odd cycle of length at least 5 or the complement of such a cycle (i.e., a complete graph minus an odd cycle).

Wagner's theorem is a result in graph theory that provides a characterization of planar graphs. Specifically, it states that a graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph \( K_{5} \) (the complete graph on five vertices) or a subdivision of the complete bipartite graph \( K_{3,3} \) (the complete bipartite graph with three vertices in each part).