Directed graphs

A directed graph (or digraph) is a type of graph in which the edges have a specific direction. This means that each edge connects an ordered pair of vertices (or nodes), indicating a one-way relationship between them. In more formal terms, if there is a directed edge from vertex \( A \) to vertex \( B \), it is often represented as \( A \rightarrow B \).

Causal diagrams are graphical representations used to illustrate relationships between variables in a system, helping to clarify the causal structures and interactions among them. They serve as a tool in fields such as statistics, epidemiology, social sciences, and causal inference to model and understand causal relationships. There are a few key types of causal diagrams: 1. **Directed Acyclic Graphs (DAGs)**: These are the most common form of causal diagrams.

A Directed Acyclic Graph (DAG) is a type of graph that has two main characteristics: 1. **Directed**: This means that the edges (connections) between the nodes (vertices) have a direction. In other words, if there is a directed edge from node A to node B, this implies a one-way relationship, indicating that A influences B, or A precedes B in some context.

Aczel's anti-foundation axiom is an alternative to the standard foundation axiom in set theory, which states that every non-empty set must contain an element that is disjoint from itself. The foundation axiom helps to avoid certain paradoxes and ensures that sets are constructed in a well-defined manner, typically preventing sets from containing themselves directly or indirectly. Aczel's anti-foundation axiom, on the other hand, allows for the existence of "non-well-founded" sets.

A **dependency graph** is a directed graph that represents dependencies between a set of items, where nodes represent the items and directed edges indicate a dependency from one item to another. In other words, if item A has a directed edge to item B, it means that A depends on B, or A cannot be completed until B has been completed.

As of my last knowledge update in October 2023, "Dicut" does not refer to a widely recognized concept, product, or term in common usage. It's possible that it could refer to a brand, a specific project, or a term used in a niche context that has emerged more recently.

As of my last knowledge update in October 2023, "Dijoin" does not refer to a widely recognized concept, product, or service in popular culture, technology, or other fields. It’s possible that it could be a term that has emerged in a specific niche or context after that date, or it might refer to a smaller-scale or localized project or organization.

A Directed Acyclic Graph (DAG) is a type of graph that has the following characteristics: 1. **Directed**: The edges in the graph have a direction, meaning that they point from one vertex (or node) to another. This is often represented with arrows on the edges.

A Kautz graph is a type of directed graph that is used in combinatorial design and graph theory. It is defined by a specific set of parameters that determine its structure. The Kautz graph \( K(n, k) \) is constructed using two parameters: \( n \) and \( k \).

The Lucchesi–Younger theorem is a result in the field of combinatorial optimization, particularly related to the study of directed graphs and their networks. The theorem states that for any directed acyclic graph (DAG), there exists a way to assign capacities to the edges of the graph such that the maximum flow from a designated source node to a designated sink node can be achieved by the flow through a certain subset of the edges.

The New Digraph Reconstruction Conjecture is a conjecture in graph theory, specifically concerning directed graphs (digraphs). It builds upon the classical Reconstruction Conjecture concerning simple (undirected) graphs. The classical Reconstruction Conjecture posits that a graph with at least three vertices can be uniquely reconstructed (up to isomorphism) from the collection of its vertex-deleted subgraphs.

The Random Surfing Model is a mathematical framework used primarily to understand and analyze the behavior of users navigating through a network, often in the context of the internet or web pages. The model simulates the process of users randomly selecting links to traverse from one node (or webpage) to another, emulating how individuals may navigate through a vast network.

**St-connectivity** refers to a concept in graph theory, particularly in the context of directed and undirected graphs. It concerns whether there is a path between two specific vertices in a graph, typically denoted as vertex **S** and vertex **T**: 1. **In Undirected Graphs**: A graph is said to be **st-connected** if there exists a path between vertices **S** and **T**.

A **strongly connected component** (SCC) is a concept from graph theory, specifically in the study of directed graphs (digraphs). In a directed graph, a strongly connected component is defined as a maximal subgraph in which every pair of vertices is reachable from each other.

The term "vertex-symmetric digraphs" typically refers to directed graphs (digraphs) that exhibit a certain level of symmetry with respect to their vertices. In general, a directed graph consists of vertices and directed edges (arcs) connecting them, and a vertex-symmetric digraph is one that behaves the same way when vertices are permuted.

In graph theory, a **tournament** is a special type of directed graph (digraph) that represents the outcomes of pairwise competitions among a set of participants. Specifically, a tournament consists of a finite set of vertices, where each vertex represents a participant, and for every pair of distinct vertices \(u\) and \(v\), there is exactly one directed edge.

A transpose graph (or the transpose of a directed graph) is a graph obtained by reversing the direction of all the edges in the original graph. In other words, if the original directed graph has an edge from vertex A to vertex B, the transpose graph will have an edge from vertex B to vertex A.

A Wait-for graph is a type of directed graph used in computer science, particularly in the context of database management systems and concurrent programming, to represent the wait-for relationships between transactions or processes. It is primarily used to detect deadlocks. In a Wait-for graph: - Each node represents a transaction or process. - A directed edge from node A to node B indicates that transaction A is waiting for a resource that is currently held by transaction B.

Why–because analysis is a causal analysis technique used to identify the root causes of problems or events. It is a structured approach that helps teams break down complex issues into simpler components by asking "why" repeatedly to delve deeper into the reasons behind a particular outcome, and then explaining that reasoning by stating "because." The purpose of this analysis is to understand the relationship between causes and effects in order to identify and address underlying issues.

Woodall's conjecture is a statement in number theory related to prime numbers. It posits that for every positive integer \( n \), there exists a prime number that can be expressed in the form \( n \cdot 2^n - 1 \). The conjecture is named after mathematician Richard Woodall, who suggested it in the context of prime-generating formulas.