Planar graphIn graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
Graph theoryIn mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.
Line graphIn the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G). The name line graph comes from a paper by although both and used the construction before this.
Graph partitionIn mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the groups will produce edges in the partitioned graph. If the number of resulting edges is small compared to the original graph, then the partitioned graph may be better suited for analysis and problem-solving than the original.
Chordal graphIn the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree.
Graph automorphismIn the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity. Formally, an automorphism of a graph G = (V, E) is a permutation σ of the vertex set V, such that the pair of vertices (u, v) form an edge if and only if the pair (σ(u), σ(v)) also form an edge. That is, it is a graph isomorphism from G to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs.
Graph isomorphism problemThe graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational complexity class NP-intermediate. It is known that the graph isomorphism problem is in the low hierarchy of class NP, which implies that it is not NP-complete unless the polynomial time hierarchy collapses to its second level.
Directed acyclic graphIn mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called arcs), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge directions.
Complement graphIn the mathematical field of graph theory, the complement or inverse of a graph G is a graph H on the same vertices such that two distinct vertices of H are adjacent if and only if they are not adjacent in G. That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there. The complement is not the set complement of the graph; only the edges are complemented. Let G = (V, E) be a simple graph and let K consist of all 2-element subsets of V.
Null graphIn the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph"). The order-zero graph, K_0, is the unique graph having no vertices (hence its order is zero). It follows that K_0 also has no edges. Thus the null graph is a regular graph of degree zero. Some authors exclude K_0 from consideration as a graph (either by definition, or more simply as a matter of convenience).
Polynomial-time reductionIn computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming or reducing it to inputs for the second problem and calling the subroutine one or more times. If both the time required to transform the first problem to the second, and the number of times the subroutine is called is polynomial, then the first problem is polynomial-time reducible to the second.
Graph (discrete mathematics)In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges.
Small-world networkA small-world network is a mathematical graph in which most nodes are not neighbors of one another, but the neighbors of any given node are likely to be neighbors of each other. Due to this, most neighboring nodes can be reached from every other node by a small number of hops or steps. Specifically, a small-world network is defined to be a network where the typical distance L between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes N in the network, that is: while the global clustering coefficient is not small.
Lattice graphIn graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space \mathbb{R}^n, forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid.
Directed graphIn mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. In formal terms, a directed graph is an ordered pair where V is a set whose elements are called vertices, nodes, or points; A is a set of ordered pairs of vertices, called arcs, directed edges (sometimes simply edges with the corresponding set named E instead of A), arrows, or directed lines.
Graph rewritingIn computer science, graph transformation, or graph rewriting, concerns the technique of creating a new graph out of an original graph algorithmically. It has numerous applications, ranging from software engineering (software construction and also software verification) to layout algorithms and picture generation. Graph transformations can be used as a computation abstraction. The basic idea is that if the state of a computation can be represented as a graph, further steps in that computation can then be represented as transformation rules on that graph.
Transitive reductionIn the mathematical field of graph theory, a transitive reduction of a directed graph D is another directed graph with the same vertices and as few edges as possible, such that for all pairs of vertices v, w a (directed) path from v to w in D exists if and only if such a path exists in the reduction. Transitive reductions were introduced by , who provided tight bounds on the computational complexity of constructing them. More technically, the reduction is a directed graph that has the same reachability relation as D.
Graph databaseA graph database (GDB) is a database that uses graph structures for semantic queries with nodes, edges, and properties to represent and store data. A key concept of the system is the graph (or edge or relationship). The graph relates the data items in the store to a collection of nodes and edges, the edges representing the relationships between the nodes. The relationships allow data in the store to be linked together directly and, in many cases, retrieved with one operation.
Spectral clusteringIn multivariate statistics, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity matrix is provided as an input and consists of a quantitative assessment of the relative similarity of each pair of points in the dataset. In application to image segmentation, spectral clustering is known as segmentation-based object categorization.
Graph minorIn graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges, vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K5 nor the complete bipartite graph K3,3. The Robertson–Seymour theorem implies that an analogous forbidden minor characterization exists for every property of graphs that is preserved by deletions and edge contractions.
