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Directed graph representing a Boolean expression
In mathematical logic and graph theory, an implication graph is a skew-symmetric, directed graph G = (V, E) composed of vertex set V and directed edge
Implication_graph
Logic problem, AND of pairwise ORs
type of directed graph, the implication graph, which expresses the variables of an instance and their negations as vertices in a graph, and constraints
2-satisfiability
Topics referred to by the same term
other Implication table, a tool used to facilitate the minimization of states in a state machine Implication graph, a skew-symmetric directed graph used
Implication
SAT solving algorithm
resulting graph is called an implication graph. Arbitrarily pick another branching variable, x3. Apply unit propagation and find the new implication graph. Here
Conflict-driven clause learning
Conflict-driven_clause_learning
Directed graph isomorphic to its own transpose graph
the implication graphs used to efficiently solve the 2-satisfiability problem. As defined, e.g., by Goldberg & Karzanov (1996), a skew-symmetric graph G
Skew-symmetric_graph
Unrelated vertices in graphs
finding a maximum independent set of a graph. Every maximum independent set also is maximal, but the converse implication does not necessarily hold. A set is
Independent set (graph theory)
Independent_set_(graph_theory)
Area of discrete mathematics
computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context
Graph_theory
Type of knowledge base
knowledge graph is a knowledge base that uses a graph-structured data model or topology to represent and operate on data. Knowledge graphs are often used
Knowledge_graph
Class of artificial neural networks
Graph neural networks (GNNs) are artificial neural networks designed for tasks whose inputs are graphs. Because graphs usually do not have a canonical
Graph_neural_network
Logical connective
implication) is a binary operation commonly used in logic. When the conditional symbol → {\displaystyle \to } is interpreted as material implication,
Material_conditional
Partition of a graph whose components are reachable from all vertices
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly
Strongly_connected_component
Directed graph with reversed edges
taking the converse of an implication in logic. The name transpose is because the adjacency matrix of the transpose directed graph is the transpose of the
Transpose_graph
Type of search algorithm
green clauses becomes True After making several decisions, we find an implication graph that leads to a conflict. Now backtrack to immediate level and by
DPLL_algorithm
Graphical representation of energy flows in physical systems
A bond graph is a graphical representation of the energy flows through, and between physical dynamical systems including those in the electrical, mechanical
Bond_graph
Generalization of graph theory
hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two
Hypergraph
On coloring graphs without complete minors
the implication, the conjecture can equivalently be stated in the following form. According to it, if all proper colorings of an undirected graph G use
Hadwiger conjecture (graph theory)
Hadwiger_conjecture_(graph_theory)
Graph representing social relations
social graph is a graph that represents social relations between entities. It is a model or representation of a social network. The social graph has been
Social_graph
Two closely related models for generating random graphs
the mathematical field of graph theory, the Erdős–Rényi models are two closely related models for generating random graphs and the evolution of a random
Erdős–Rényi_model
Knowledge management software
knowledge elements can be structured and interrelated (as a tree, graph, tree plus graph, spatially, categorically, as n-ary links, chronologically, or ZigZag);
Personal_knowledge_base
Result in combinatorics and graph theory
equivalent for a bipartite graph G = (X+Y, E): G admits an X-perfect matching. G admits an X-perfect fractional matching. The implication follows directly from
Hall's_marriage_theorem
Unproven computational hardness assumption
problems include graph k-colorability, finding Hamiltonian cycles, maximum cliques, maximum independent sets, and vertex cover on n-vertex graphs. Conversely
Exponential_time_hypothesis
Mathematically obvious
material implication P→Q, where the consequent Q, is always true. Here, the proof follows immediately by virtue of the definition of material implication in
Triviality_(mathematics)
Logical formulation of graph properties
the mathematical fields of graph theory and finite model theory, the logic of graphs deals with formal specifications of graph properties using sentences
Logic_of_graphs
Equivalence of distributive lattices and set families
lower sets of the transitive closure of the graph. Equivalently, for a distributive lattice, the implication graph of the 2-satisfiability instance can be
Birkhoff's representation theorem
Birkhoff's_representation_theorem
Fair division protocol
the implication graph are dependent. but thanks to the semi-final selection phase, we can prove that the probability that the implication graph contains
Edmonds–Pruhs_protocol
Directed graph that models causal relationships between variables
inference tools, the graphs enable researchers to estimate effect sizes from non-experimental data, derive testable implications of the assumptions encoded
Causal_graph
On forbidden subgraphs in planar graphs
In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states
Kuratowski's_theorem
In logic, a statement which is always true
{\displaystyle R} to be true, and so the definition of tautological implication is trivially satisfied. Similarly, if S {\displaystyle S} is a tautology
Tautology_(logic)
Fuzzy logic concept
interpreted by a t-norm, the residuum plays the role of implication (often called R-implication). If ⇒ {\displaystyle \Rightarrow } is the residuum of
T-norm
Computational Formula that can be measured in terms of True or False
time, by an algorithm involving strong connectivity analysis of its implication graph. The 2-satisfiability problem is a special case of TQBF for these
True quantified Boolean formula
True_quantified_Boolean_formula
Graph in climate science
Hockey stick graphs present global or hemispherical mean global surface temperatures of the distant past, as shown by quantitative climate reconstructions
Hockey stick graph (global temperature)
Hockey_stick_graph_(global_temperature)
Clustering methods
k-means in the eigenspace defined by the graph Laplacian. This theoretical insight has practical implications: the final clustering step in spectral clustering
Spectral_clustering
Structuring text as input to generative artificial intelligence
frequent retraining. GraphRAG (coined by Microsoft Research) is a technique that extends RAG with the use of a knowledge graph to allow the model to
Prompt_engineering
Every graph has evenly many odd vertices
In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges
Handshaking_lemma
Relationship in which one statement follows from another
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that
Logical_consequence
Measure of the structural complexity of a software program
Cyclomatic complexity is computed using the control-flow graph of the program. The nodes of the graph correspond to indivisible groups of commands of a program
Cyclomatic_complexity
Theorem of convex functions
the statement that the secant line of a convex function lies above the graph of the function, which is Jensen's inequality for two points: the secant
Jensen's_inequality
Statement in mathematical combinatorics
its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. As
Ramsey's_theorem
Graph literacy is the ability to understand information that presented graphically, which are including general knowledge about how to extract information
Graph_literacy
Theorem in graph theory
pairwise edge-disjoint paths from x to y. The implication for the graph G is the following version: A graph is k-edge-connected (it remains connected after
Menger's_theorem
Open problem on 3x+1 and x/2 functions
13, 40, 20, 10, 5, 16, 8, 4, 2, 1 . The sequence for n = 27, listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing
Collatz_conjecture
Graph obeys some properties of random graphs
In graph theory, a graph is said to be a pseudorandom graph if it obeys certain properties that random graphs obey with high probability. There is no concrete
Pseudorandom_graph
On graphs with given symmetry groups
symmetries of an infinite graph; alternatively, the axiom of regularity eliminates the need for choice. Furthermore, this implication is strict, as there are
Frucht's_theorem
On coloring the edges of graphs
In graph theory, Vizing's theorem states that every simple undirected graph may be edge colored using a number of colors that is at most one larger than
Vizing's_theorem
Subset of a graph's nodes such that all other nodes link to at least one
In graph theory, a dominating set for a graph G is a subset D of its vertices, such that any vertex of G is in D, or has a neighbor in D. The domination
Dominating_set
Mathematical concept
the graph of the function y=f−1(x) is the same as the graph of the equation x=f(y). This is identical to the equation y = f(x) that defines the graph of
Inverse_function
Process by which people befriend similar people
policies have a decreased influence on fertility rates in such populations. In graph representation learning, homophily means that nodes with the same label
Homophily
Binary relation over a set and itself
endorelations. Terminology particular for graph theory is used for description, with an ordinary (undirected) graph presumed to correspond to a symmetric
Homogeneous_relation
Measure of a graph's centrality, based on shortest paths
In graph theory, betweenness centrality is a measure of centrality in a graph based on shortest paths. Betweenness centrality measures how frequently a
Betweenness_centrality
Property of a mathematical operation
Transformation rules Propositional calculus Rules of inference (List) Implication introduction / elimination (modus ponens) Biconditional introduction /
Associative_property
Mathematical theory on behavior of connected clusters in a random graph
bottom? Similarly, one can ask, given a connected graph at what fraction 1 – p of failures the graph will become disconnected (no large component). The
Percolation_theory
Finiteness of sets of forbidden graph minors
graph theory, the Robertson–Seymour theorem (also called the graph minors theorem) states that the undirected graphs, partially ordered by the graph minor
Robertson–Seymour_theorem
path between two given vertices in a weighted graph, finding negative-weight triangles in weighted graphs, and testing whether a given distance matrix
Fine-grained_reduction
conjecture, if true, would have numerous implications in graph theory, including the 1-factorization conjecture. For graphs in which Δ ≥ n / 3 {\displaystyle
Overfull_graph
Group that admits a formal description in terms of reflections
(yx)^{m}=(yx)^{m}yy=y(xy)^{m}y=yy=1} . An alternative proof of this implication is the observation that ( x y ) k {\displaystyle (xy)^{k}} and ( y x
Coxeter_group
Terms to describe a conditional relationship between two statements
necessity and sufficiency are concepts used to denote a conditional or implicational relationship between two statements. For example, in the conditional
Necessity_and_sufficiency
Search engine from Google
the feature expanded to all countries globally in 2017. The Knowledge Graph is a knowledge base used by Google to enhance its search engine's results
Google_Search
Concept in mathematical logic
converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication P → Q, the converse is Q
Converse_(logic)
Analysing a string of symbols, according to the rules of a formal grammar
languages with layered graph grammars." Journal of Visual Languages & Computing 8.1 (1997): 27-55. Rekers, Jan, and A. Schurr. "A graph grammar approach to
Parsing
Deviation from equilibrium price and quantity
Graph of a tax wedge, showing consumer and producer incidence.
Tax_wedge
On short connecting nets with added points
term Steiner tree problem, is the Steiner tree problem in graphs. Given an undirected graph with non-negative edge weights and a subset of vertices, usually
Steiner_tree_problem
Inherent difficulty of computational problems
is the following. The input is an arbitrary graph. The problem consists in deciding whether the given graph is connected or not. The formal language associated
Computational complexity theory
Computational_complexity_theory
Continuous function on an interval takes on every value between its values at the ends
3 , f ( 2 ) = 5 {\displaystyle f\in C([1,2]),f(1)=3,f(2)=5} , then the graph of y = f ( x ) {\displaystyle y=f(x)} must pass through the horizontal line
Intermediate_value_theorem
Network whose degree distribution follows a power law
Willinger, W. (2005). "Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)". arXiv:cond-mat/0501169. Kumar
Scale-free_network
In statistics and Markov modeling, an ancestral graph is a type of mixed graph used to provide a graphical representation for the result of marginalizing
Ancestral_graph
Mathematical model as an economic concept
James Choi states that this deviates from the advice of economists. The graph below illustrates the expected utility model, in which U(c) is increasing
Consumption_smoothing
NP-hard problem in combinatorial optimization
version of the TSP (where given a length L, the task is to decide whether the graph has a tour whose length is at most L) belongs to the class of NP-complete
Travelling_salesman_problem
bipartite graphs. Degree matrix — a diagonal matrix defining the degree of each vertex in a graph. Edmonds matrix — a square matrix of a bipartite graph. Incidence
List_of_named_matrices
Task of computing complete subgraphs
vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several different formulations depending on which cliques, and what
Clique_problem
Unsolved problem in computer science
(2006). "Graph isomorphism is in SPP". Information and Computation. 204 (5): 835–852. doi:10.1016/j.ic.2006.02.002. Schöning, Uwe (1988). "Graph isomorphism
P_versus_NP_problem
French mathematician (1875–1941)
operation that corresponds to the informal idea of finding the area under the graph of a function. The first theory of integration was developed by Archimedes
Henri_Lebesgue
Visual representation of the structure of an argument
"arrows link reasons and conclusions (but) no support is given to the implication itself between them. There is no theory, in other words, of inference
Argument_map
biconditional Logical conjunction Logical disjunction Logical equality Logical implication Logical negation Logical NOR Majority function Material conditional Minimal
List of Boolean algebra topics
List_of_Boolean_algebra_topics
Security feature of macOS
Calendar Contacts Control Center Dictionary FaceTime Finder Game Center Grapher Home Mail Messages News Music Notes Notification Center Podcasts Photo
Gatekeeper_(macOS)
Planar maps require at most four colors
terms of graph theory, by considering it in terms of constructing a graph coloring of the planar graph of adjacencies between regions. In graph-theoretic
Four_color_theorem
2017 German science-fiction television series
that spans several generations. The series explores the existential implications of time and its effect on human nature and life, with its plot structure
Dark_(TV_series)
Length of time that copyright continues
Term (July 23, 2008) Clorox (diskussion)'s Vectorization of Tom Bell's graph, depicted above, which shows expansion of U.S. copyright law (November 27
Copyright_term
Obsolete unit of account of the European Community
Retrieved 19 August 2020. Economic and Monetary Union at the European Central Bank Google Ngrams: Historical usage graph of "ecus" and "euros" in English
European_Currency_Unit
Relation between surface area and volume as size increases
surface area. When applied to the real world, this principle has many implications which are important in fields ranging from mechanical engineering to
Square–cube_law
On forbidden minors in planar graphs
In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite
Wagner's_theorem
Graph drawing with vertices on a line
An arc diagram is a style of graph drawing, in which the vertices of a graph are placed along a line in the Euclidean plane and edges are drawn using
Arc_diagram
Graphical set representation involving overlapping shapes
discusses the fact that the implication P → Q need not be a tautology (a so-called "tautological implication"). Even "simple" implication (connective or adjunctive)
Euler_diagram
Graph in which all neighbors of every vertex have distinct degrees
In graph theory, a highly irregular graph is a graph in which, for every vertex, all neighbors of that vertex have distinct degrees. Irregular graphs were
Highly_irregular_graph
Pan-genome Graph Construction Methodology
Pan-genome graph construction is the process of creating a graph-based representation of the collective genome (the pan-genome) of a species or a group
Pan-genome_graph_construction
Format for expressing RDF statements in HTML documents
annotate data in web pages Open Graph protocol, a way to use RDFa to integrate web pages into the Facebook social graph Schema.org, search-engine supported
RDFa
Algebraic structure used in logic
and greatest element 1) equipped with a binary operation a → b called implication such that (c ∧ a) ≤ b is equivalent to c ≤ (a → b). In a Heyting algebra
Heyting_algebra
Professional network website
analytics Inspired by Facebook's "social graph", LinkedIn CEO Jeff Weiner set a goal in 2012 to create an "economic graph" within a decade. The goal was to create
Concept in computer science
− ∗ {\displaystyle -\!\!\ast } (pronounced magic wand or separating implication) asserts that extending the heap with a disjoint part that satisfies
Separation_logic
Overview and analysis of structured formats in journalism
by the methodology used, and conclude with implications and future research directions. In the nut graph structure, a short paragraph provides the context
Article_structure
Proposition that visual inscriptions are central to scientific practice
presentation, with tables occupying more space than graphs. The graphism thesis has implications for several areas within the philosophy of science and
Graphism_thesis
Square matrix without an inverse
or apply forces in certain directions. In graph theory and network physics, the Laplacian matrix of a graph is inherently singular (it has a zero eigenvalue)
Singular_matrix
Mathematical model of ferromagnetism in statistical mechanics
that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all
Ising_model
Study of the attraction between people that leads to friendship or romance
individual human person as pleasing, or beautiful. It can include various implications, such as sexual attractiveness, cuteness, similarity and physique. Judgment
Interpersonal_attraction
Type of computer science algorithm
one wishes to know if two vertices in a graph of n vertices are in the same connected component of the graph, there is no known simple, deterministic
In-place_algorithm
Journalistic adage on questions in headlines
deny facts, such as "Is Venkaiah Naidu on his way out?" Because this implication is known to readers, guides giving advice to newspaper editors state
Betteridge's_law_of_headlines
Computational problem in graph theory
In graph theory and combinatorial optimization, a closure of a directed graph is a set of vertices C, such that no edges leave C. The closure problem is
Closure_problem
Conjecture in probabilistic combinatorics
mathematics that studies the behavior of connected clusters in a random graph. The conjecture is named after its analogy to a bunk bed structure. It was
Bunkbed_conjecture
-dimensional normed vector space is a property of graphs which states that any embedding, or drawing, of the graph in some high dimension d ′ {\displaystyle d'}
Graph_flattenability
Subfield of computational complexity theory
Hundreds of problems across computational geometry, string algorithms, dynamic graph algorithms, and data structures have been shown hard under one or more of
Fine_grained_complexity
Mars 2020 electrochemical experiment
MOXIE first martian oxygen production test on April 20, 2021, graph
Mars_Oxygen_ISRU_Experiment
travel, tourism, insurance
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IMPLICATION GRAPH
travel, tourism, insurance