Vertex (graph theory)In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another.
Complete graphIn the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull.
Shallow minorIn graph theory, a shallow minor or limited-depth minor is a restricted form of a graph minor in which the subgraphs that are contracted to form the minor have small diameter. Shallow minors were introduced by , who attributed their invention to Charles E. Leiserson and Sivan Toledo. One way of defining a minor of an undirected graph G is by specifying a subgraph H of G, and a collection of disjoint subsets Si of the vertices of G, each of which forms a connected induced subgraph Hi of H.
Coxeter graphIn the mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. It is one of the 13 known cubic distance-regular graphs. It is named after Harold Scott MacDonald Coxeter. The Coxeter graph has chromatic number 3, chromatic index 3, radius 4, diameter 4 and girth 7. It is also a 3-vertex-connected graph and a 3-edge-connected graph. It has book thickness 3 and queue number 2. The Coxeter graph is hypohamiltonian: it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian.
Disjoint-set data structureIn computer science, a disjoint-set data structure, also called a union–find data structure or merge–find set, is a data structure that stores a collection of disjoint (non-overlapping) sets. Equivalently, it stores a partition of a set into disjoint subsets. It provides operations for adding new sets, merging sets (replacing them by their union), and finding a representative member of a set. The last operation makes it possible to find out efficiently if any two elements are in the same or different sets.
Polygonal chainIn geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain P is a curve specified by a sequence of points called its vertices. The curve itself consists of the line segments connecting the consecutive vertices. A simple polygonal chain is one in which only consecutive segments intersect and only at their endpoints. A closed polygonal chain is one in which the first vertex coincides with the last one, or, alternatively, the first and the last vertices are also connected by a line segment.
Pappus graphIn the mathematical field of graph theory, the Pappus graph is a bipartite 3-regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek mathematician who is believed to have discovered the "hexagon theorem" describing the Pappus configuration. All the cubic distance-regular graphs are known; the Pappus graph is one of the 13 such graphs. The Pappus graph has rectilinear crossing number 5, and is the smallest cubic graph with that crossing number .
BisectionIn geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a 'bisector'. The most often considered types of bisectors are the 'segment bisector' (a line that passes through the midpoint of a given segment) and the 'angle bisector' (a line that passes through the apex of an angle, that divides it into two equal angles). In three-dimensional space, bisection is usually done by a bisecting plane, also called the 'bisector'.
Vertex configurationIn geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.) A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "a.
AsymptoteIn analytic geometry, an asymptote (ˈæsɪmptoʊt) of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (asumptōtos) which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen".
