Graph theory is a branch of mathematics that studies graphs, which are mathematical structures used to model pairwise relationships between objects. In graph theory, the objects can be represented in various ways, but the fundamental components include: 1. **Vertices (Nodes)**: These are the fundamental units of a graph that represent the entities or objects. For example, in a social network, vertices could represent people. 2. **Edges (Links)**: These are the connections between pairs of vertices.
A **Hamiltonian path** is a specific type of path in a graph that visits each vertex exactly once. In other words, it is a trail in which every node (or vertex) of the graph is included exactly one time. A **Hamiltonian cycle** (or Hamiltonian circuit) is a special case of a Hamiltonian path where the path starts and ends at the same vertex, thus forming a closed loop that visits every vertex once.
Barnette's conjecture is a proposition in the field of combinatorial geometry, specifically concerning polyhedra. It states that for a polyhedron with \( n \) vertices, the number of faces \( f \) must satisfy the inequality: \[ f \leq 2n - 4 \] This conjecture essentially posits an upper bound on the number of faces in a convex polyhedron based on its number of vertices.
Hamiltonian completion is a concept in graph theory related to the idea of completing a given graph into a Hamiltonian graph. A Hamiltonian graph is one that contains a Hamiltonian cycle, which is a cycle that visits every vertex in the graph exactly once and returns to the starting vertex. Hamiltonian completion specifically deals with taking an incomplete graph (one that may not be Hamiltonian) and determining whether it is possible to add a certain number of edges to make it Hamiltonian.
The Herschel graph, also known as the Herschel-Dickson graph, is a specific type of undirected graph that is notable in the study of mathematical graphs and combinatorial design. It is a bipartite graph that is defined as follows: 1. **Vertices**: The Herschel graph consists of 14 vertices. It can be visualized as having two sets of vertices: - One set consists of 7 vertices (usually denoted as \( U \)).
A hypohamiltonian graph is a type of graph in graph theory that is defined as follows: a graph \( G \) is considered hypohamiltonian if it is not Hamiltonian (i.e., it does not contain a Hamiltonian circuit) but the removal of any single vertex from \( G \) results in a graph that is Hamiltonian.
A **pancyclic graph** is a type of graph in graph theory that contains cycles of all possible lengths from 3 up to the maximum length that is less than or equal to the number of vertices in the graph.
The term "shortness exponent" isn't widely known or defined within established scientific literature as of my last update. However, it's possible that it may refer to a concept in a specialized area of research, possibly in fields like physics, mathematics, or data analysis, where exponents are used to characterize statistical properties of distributions or phenomena. If you're referring to a concept in a specific context (e.g.
A **subhamiltonian graph** is a type of graph in the field of graph theory. Specifically, a subhamiltonian graph is one that contains a Hamiltonian path but not necessarily a Hamiltonian cycle. In other words, it is possible to traverse all vertices in the graph exactly once (the definition of a Hamiltonian path), but it may not be possible to return to the starting vertex without repeating any vertices (which would be needed for a Hamiltonian cycle).
Tait's conjecture is a statement in graph theory and topology related to the study of knot diagrams. Proposed by the Scottish mathematician Peter Tait in the 19th century, the conjecture specifically pertains to the number of crossings in alternating knot diagrams.
A Walther graph is a type of graph that arises in the context of graph theory, particularly in the study of order types and combinatorial structures. It is constructed using the points of a finite projective plane. Specifically, a Walther graph is formed from a set of points and lines in a projective plane, where the vertices of the graph represent points, and edges connect pairs of vertices if the corresponding points lie on the same line.
An acyclic orientation of a directed graph (digraph) is an assignment of directions to the edges of the graph such that there are no directed cycles.
The term "bipolar orientation" typically refers to a sexual or romantic orientation characterized by attraction to individuals of two or more genders. However, it's important to clarify that the more commonly used term for this orientation is "bisexual." Bisexuality encompasses a range of experiences and identities, and individuals may identify as bisexual in different ways, reflecting their unique attractions and experiences.
In graph theory, a **Blossom** refers to a specific structure that is relevant in the context of matching algorithms, particularly in the matching of general graphs. The Blossom structure is utilized to handle situations where augmenting paths may be of odd length, which can complicate the process of finding maximum matchings. The concept of Blossoms is associated with the **Edmonds' Blossom Algorithm**, developed by Jack Edmonds in the 1960s.
Chordal completion, also known as chordal graph completion, refers to the process of transforming a given graph into a chordal graph. A chordal graph (or "perfect graph") is defined as a graph in which every cycle of four or more vertices has a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. The process of chordal completion involves adding edges to the original graph in such a way that the resulting graph becomes chordal.
In graph theory, a **clique** is a subset of vertices in an undirected graph such that every two distinct vertices in the clique are adjacent. In simpler terms, a clique is a complete subgraph where every vertex is connected to every other vertex. More formally, a clique of size \( k \) is a set of \( k \) vertices such that there are edges between every pair of vertices in that set.
In graph theory, the term "core" refers to a specific type of subgraph that captures some essential structural properties of the original graph. A **core** of a graph is often defined as a maximal subgraph in which every vertex has a degree (number of edges connected to it) of at least \( k \). This means that for a \( k \)-core, every vertex in the graph has at least \( k \) connections.
In graph theory, a **cycle** is a path that starts and ends at the same vertex, with all other vertices in the path being distinct. More formally, a cycle in an undirected graph is defined as a sequence of vertices \( v_1, v_2, ...
Edge-graceful labeling is a concept in graph theory related to the labeling of the edges of a graph. Specifically, it refers to a particular way of assigning labels from a set of integers to the edges of a graph such that certain conditions are met.
An edge cycle cover in graph theory is a collection of cycles that covers every edge of a graph exactly once. In other words, it is a set of cycles in which each edge of the graph is included in exactly one of the cycles. This concept is particularly relevant in the study of Eulerian paths and circuits.
In graph theory, an "end" refers to a concept that is used to describe the behavior of infinite graphs. More formally, an end is a way of capturing the idea of "directions" or "ways to escape" from a finite portion of a graph toward infinity.
An **eternal dominating set** is a concept from graph theory, particularly in the study of domination in graphs. The idea revolves around the ability to monitor or control the vertices of a graph over time, adapting to changes such as the removal of vertices.
An Eulerian path is a trail in a graph that visits every edge exactly once. It can begin and end at different vertices. The concept is named after the Swiss mathematician Leonhard Euler, who introduced it in the context of solving the famous Seven Bridges of Königsberg problem.
Graceful labeling is a concept in graph theory related to labeling the vertices of a graph in a specific way that satisfies certain criteria. A graph is said to be gracefully labeled if it can be assigned labels (usually integers) to its vertices such that the following conditions are met: 1. The labels assigned to the vertices are distinct integers, typically taken from the set {0, 1, 2, ...
In graph theory, the **center** of a graph is a concept that refers to a specific set of vertices that minimize the maximum distance to all other vertices in the graph. In other words, the center of a graph consists of those vertices from which the farthest distance to any other vertex in the graph is minimized.
Hamiltonian decomposition is a concept in graph theory, particularly concerned with the decomposition of graphs into Hamiltonian cycles or paths. A **Hamiltonian cycle** is a cycle that visits every vertex of a graph exactly once and returns to the starting vertex, while a **Hamiltonian path** visits every vertex exactly once but does not return to the starting vertex. In Hamiltonian decomposition, the objective is to represent a given graph as a collection of Hamiltonian cycles or paths.
In graph theory, the term "incidence" refers to the relationship between the edges and the vertices of a graph. Specifically, it describes how edges connect to vertices. In a graph: - A **vertex** is a point or a node. - An **edge** is a line connecting two vertices. There are a few important concepts associated with incidence in graphs: 1. **Incidence Relation**: An edge is said to be incident to the vertices it connects.
The term "induced path" typically arises in various contexts, such as in mathematics, particularly in graph theory. In graph theory, an **induced path** refers to a specific kind of subgraph of a graph.
The term "level structure" can refer to various concepts depending on the context. Here are a few interpretations: 1. **Education**: In educational contexts, level structure might refer to the organization of courses and programs into different levels (e.g., introductory, intermediate, advanced) to help guide students through their learning journey. 2. **Video Games**: In video game design, level structure refers to how different levels or stages of a game are organized and designed.
In graph theory, a "map" typically refers to a representation of a geographical area or a network of connections that can be modeled using graphs. This involves vertices (or nodes) and edges (or links) that represent different entities and their relationships. In one common interpretation, a map can refer to a planar graph, which is a graph that can be embedded in the plane without any edges crossing each other.
In the context of graph theory, a **minimum cut** refers to a specific type of partitioning of a graph that separates the vertices into two distinct subsets while minimizing the total weight of the edges that cross between these two subsets. ### Key Concepts: - **Graph**: A collection of vertices (nodes) connected by edges (links). - **Cut**: A cut in a graph is a way of dividing the graph's vertices into two disjoint subsets.
Modular decomposition is a concept primarily used in graph theory and computer science, particularly in the study of algorithms and structures related to graphs. It involves breaking down a graph into its simpler, modular components or modules based on the relationships between its vertices. ### Key Concepts: 1. **Module**: In a graph, a module (or a strongly connected component) is a subset of vertices such that every vertex in this subset is equally connected to all the other vertices in the subset.
In graph theory, **multiple edges** refer to a situation in a graph where two vertices are connected by more than one edge. This is common in **multigraphs**, which are a type of graph that allows for multiple edges between the same pair of vertices as well as loops (edges that connect a vertex to itself).
In graph theory, a **neighbourhood** typically refers to the set of vertices that are directly adjacent (or connected) to a given vertex in a graph.
In graph theory, an "orientation" of a graph refers to a directed version of the graph obtained from an undirected graph (or simply a graph) by assigning a direction to each of its edges.
In graph theory, a **path cover** in a directed graph is a set of vertex-disjoint paths that collectively include every vertex of the graph. In other words, a path cover partitions the vertices of the graph into paths, meaning that each vertex belongs to exactly one path, and there are no overlapping vertices between different paths.
The term "peripheral cycle" can refer to different concepts depending on the context in which it is used. Here are a couple of possible interpretations: 1. **Peripheral Cycle in Finance**: In finance, a "peripheral cycle" might refer to the cyclical movements in the economic performance of peripheral economies, particularly those that are not at the center of global financial markets.
A Sachs subgraph is a concept from graph theory, specifically in the context of combinatorial structures and properties related to planar graphs. It refers to a type of subgraph that is constructed from a planar graph in such a way that certain conditions are met. In more detail, a Sachs subgraph is typically defined for a planar graph as a spanning subgraph that respects the original graph's facial structures in a certain way.
A skew partition refers to a method of organizing data that partitions a dataset in such a way that it may be unbalanced or uneven, leading to potential performance issues in distributed systems or parallel processing environments. The term "skew" suggests that the distribution of data is not uniform; some partitions contain significantly more data than others. In the context of databases or data processing frameworks, skew partitions can lead to certain tasks taking considerably longer to complete because they have to handle more data than others.
In graph theory, a **split graph** is a type of graph that can be partitioned into two disjoint sets of vertices: one set forms a clique (a complete subgraph where every pair of vertices is connected by an edge), and the other set forms an independent set (a set of vertices no two of which are adjacent).
An **unfriendly partition** is a concept that arises in the context of graph theory and combinatorics. Generally, it refers to a way of partitioning a set of elements, such as vertices in a graph, where certain pairs of elements have a restricted relationship (e.g., they cannot be grouped together in the same subset) due to specific constraints.
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