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Look up vertex in Wiktionary, the free dictionary. Vertex, vertices or vertexes may refer to: Vertex (geometry), a point where two or more curves, lines...

represented by abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link...

undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the...

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...

Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the...

K3 is not a minor of G. Any two vertices in G can be connected by a unique simple path. If G has finitely many vertices, say n of them, then the above...

rows and columns are both indexed by vertices of the graph, with a one in the cell for row i and column j when vertices i and j are adjacent, and a zero otherwise...

are attached edge-to-edge, with the property that their vertices coincide with the set of vertices of the polygon. In the case of a simple polygon with n...

its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring...

subdivision surfaces will fail when a model contains T-vertices. Frequent Mesh Problems, www.cs.princeton.edu Media related to T-vertices at Wikimedia Commons...

acyclic digraphs. A graph is formed by vertices and by edges connecting pairs of vertices, where the vertices can be any kind of object that is connected...

to any of the vertices are not written down. Let v be the vertices of the graph and describe the Hamiltonian circle along the p vertices by the edge sequence...

join the vertices to the incenters of the opposite faces are concurrent. An isogonic tetrahedron has concurrent cevians that join the vertices to the points...

16 vertices, is the convex hull of a compound of two 16-cells, with 8 vertices each, in an exact dimensional analogy to the cube, with 8 vertices, which...

A cube is a three-dimensional solid object in geometry. It has eight vertices and twelve straight edges of the same length, so that these edges form six...

of its vertices such that for every directed edge (u,v) from vertex u to vertex v, u comes before v in the ordering. For instance, the vertices of the...

which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is...

its neighboring vertices. The last two vertices, 8 and 9 at the top and bottom center of the "box-cylinder", have four connected vertices rather than five...

two vertices is a unit distance graph, as is the complete graph on three vertices (the triangle graph), but not the complete graph on four vertices. Generalizing...

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...

those vertices maps to one of the simplex's n−3 faces, and so forth, and the vertices connected to the bottom vertex map to the simplex's vertices. This...

states that every simple polygon with more than three vertices has at least two ears, vertices that can be removed from the polygon without introducing...

of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with n vertices is called Cn. The number of vertices in...

surface that can be described by its vertices (corner points), edges (line segments connecting certain pairs of vertices), faces (two-dimensional polygons)...

circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an...

of vertices are only slightly larger than their dimensions. A d {\displaystyle d} -dimensional polytope with n = d + 1 {\displaystyle n=d+1} vertices, the...

Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle, Hamiltonian circuit, vertex...

vertices. In any graph without isolated vertices the size of the minimum edge cover plus the size of a maximum matching equals the number of vertices...

four vertices. A solid figure with four faces as well as four vertices is a tetrahedron, which is the smallest possible number of faces and vertices a polyhedron...

possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph...

theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color...

graph G, two vertices u and v are called connected if G contains a path from u to v. Otherwise, they are called disconnected. If the two vertices are additionally...

be stored simply as a sequence of vertices ABCDEF. This sequence would be decoded as a set of triangles with vertices at ABC, BCD, CDE and DEF - although...

n {\displaystyle n} is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n −...

(and possibly mutable) set of vertices (also called nodes or points), together with a set of unordered pairs of these vertices for an undirected graph or...

icosahedral symmetry of order 120. The Cartesian coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations...

fraction of the vertex set, called buffer vertices, and attempt to embed the rest of the vertices. The buffer vertices are subsequently embedded by using Hall's...

Vértice 360 is a Spanish audiovisual production and distribution company. It is a division of Squirrel Media [es]. Vértice 360 (also Vértice 360º) was...

of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed...

labeled vertices is n n − 2 {\displaystyle n^{n-2}} . The formula equivalently counts the spanning trees of a complete graph with labeled vertices (sequence...

first 8 are the vertices of a 16-cell, the second 16 are the vertices of a tesseract, and those 24 vertices together are the vertices of a 24-cell. The...

may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components...

frequently differed from the numbers of vertices of the Platonic solids. No ball's knobs matched the 20 vertices of the dodecahedron, and the arrangement...

cannot be the Petersen graph. If two chords connect opposite vertices of C to vertices at distance four along C, there is again a 4-cycle. The only remaining...

colouring of the complete graph on R(r, s) vertices contains a blue clique on r vertices or a red clique on s vertices. (Here R(r, s) signifies an integer that...

because antipodal vertices may be colored the same, but all nodes between them must be different. The distance between two vertices in a graph is defined...

The blocks are attached to each other at shared vertices called cut vertices or separating vertices or articulation points. Specifically, a cut vertex...

are defined in a database, containing arrays of vertices (the coordinates of the geometrical vertices, as well as other attributes of the polygon, such...

the faces that make each polygon defined as a list of vertices, and texture vertices. Vertices are stored in a counter-clockwise order by default, making...

four or fewer vertices are planar, however, there is a graph with five vertices that is not: K5, the complete graph with five vertices. By Kuratowski's...

connecting each of the golden rectangle's vertices with a segment line. There are 12 regular icosahedron vertices, considered as the center of 12 regular...

determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle. The British flag theorem states that with vertices denoted A, B...

14 vertices (3,7)-cage: the McGee graph, 24 vertices (3,8)-cage: the Tutte–Coxeter graph, 30 vertices (3,10)-cage: the Balaban 10-cage, 70 vertices (3...

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...

number n {\displaystyle n} of vertices contains a Hamiltonian path, i.e., directed path on all n {\displaystyle n} vertices (Rédei 1934). This is shown...

the vertices are visited. Tree traversal is a special case of graph traversal. Unlike tree traversal, graph traversal may require that some vertices be...

partition the vertices of the graph into subsets that are totally ordered by reachability. They form the finest partition of the set of vertices that is totally...

Vértice (Spanish: Vertex) was a monthly Falangist magazine published in Spain, between 1937 and 1946. Its subtitle was Revista nacional de la Falange...

example, the 120-cell is a compound of 25 disjoint 24-cells (25 * 24 vertices = 600 vertices). The green integers are the number of distinct 4-polytopes above...

dual structure, wherein the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges...

An undirected graph consists of a system of vertices, and edges connecting unordered pairs of vertices. In any graph, the degree deg ⁡ ( v ) {\displaystyle...

subset D of its vertices, such that any vertex of G is in D, or has a neighbor in D. The domination number γ(G) is the number of vertices in a smallest...

to a graph’s vertices by constructing each color class one at a time. It does this by identifying a maximal independent set of vertices in the graph,...

induced by all the vertices that are connected to v by an edge (vertices that are adjacent to v), i.e., the graph composed of all the vertices adjacent to v...

represented by its skeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric...

all original vertices by the common edge length. The standard simplex or probability simplex is the k-dimensional simplex whose vertices are the k + 1...

graph theory, a nonblocker is a subset of vertices in an undirected graph, all of which are adjacent to vertices outside of the subset. Equivalently, a nonblocker...

at two of the vertices of the pentagon. Draw a circle of radius OA and center V. It intersects the original circle at two of the vertices of the pentagon...

first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal....

orientations, the dummy vertices are removed from the graph and the vertices and edges are drawn. To avoid intersections between vertices and edges, edges that...

rhombic triacontahedron because only the order-5 vertices are truncated. These 12 order-5 vertices can be truncated such that all edges are of equal...

This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron form an embedding...

to all vertices of G. Thus, finding a Hamiltonian path cannot be significantly slower (in the worst case, as a function of the number of vertices) than...

polyhedron with triangular faces, and q-valence vertices. The + symbol indicates the valence of the vertices being increased. b and c represent a subdivision...

icosahedron envelope. There are 22 exterior vertices, and 10 interior vertices. The 10 interior vertices have the convex hull of a pentagonal antiprism...

{\text{Long side}}} The eight vertices of a cube have the coordinates (±1, ±1, ±1). The coordinates of the 12 additional vertices are (0, ±(1 + h), ±(1 − h2))...

polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to the vertices of Archimedean solids, and vice versa...

A directed graph, whose underlying undirected graph is a tree (any two vertices are connected by exactly one simple path), with a distinguished root (one...

triangles and 6 squares. It also has 24 edges and 12 vertices. The Cartesian coordinates for the vertices of a cuboctahedron with edge length 2 {\displaystyle...

geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping...

selecting only vertices from the subject polygon that are on the visible side. The algorithm begins with an input list of all vertices in the subject...

vertices in the graph. Edges between two contracting vertices are sometimes removed. If v {\displaystyle v} and v ′ {\displaystyle v'} are vertices of...

counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38 vertices, were later proved minimal by Holton...

formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with n vertices can also be defined as the 1-skeleton of an (n − 1)-gonal...

by the uncolored vertices. Assign v {\displaystyle v} to the lowest color label not being used by any of its neighbors. If all vertices have been colored...

enumerate the vertices of the graph, to store data per vertex (if not in the graph data structure itself, then in some table that can use vertices as indices)...

at least 17 vertices, and any cubic counterexample must have at least 30 vertices. Markström's searches found four graphs on 24 vertices in which the...

vertices such that edges are mapped to edges and non-edges are mapped to non-edges. A graph is a vertex-transitive graph if, given any two vertices v1...

result with coinciding vertices and edges. For example, {6/2} will appear as a triangle, but can be labeled with two sets of vertices: 1–3 and 4–6. This should...

k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed. The vertex-connectivity, or just...

m edges is a labeling of its vertices with some subset of the integers from 0 to m inclusive, such that no two vertices share a label, and each edge is...

(dṓdeka) 'twelve' and γραμμῆς (grammēs) 'line') is a star polygon or compound with 12 vertices. There is one regular dodecagram polygon (with Schläfli symbol {12/5} and...

guarded. Solving the version in which guards must be placed on vertices and only vertices need to be guarded is equivalent to solving the dominating set...

graph has three vertices of degree four (the three blue vertices aligned vertically in the center of the illustration) and eight vertices of degree three...

the vertices of these curves. In the case of a triangle, the medial axis consists of three segments of the angle bisectors, connecting the vertices of...

sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns...

1, …, |V| } , where |V| is the number of vertices in the graph. For many applications, the edges or vertices are given labels that are meaningful in the...

include both vertices and edges or only vertices as first class objects, and in what kinds of objects are used to represent the vertices and edges. An...

{1}{24}}\left(m^{6}+3m^{4}+12m^{3}+8m^{2}\right)} different colorings. A graph on m vertices can be interpreted as an arrangement of colored beads. The set X of "beads"...

− 1. Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices of degree 1), while all others (if any) have...

cantellation, cuts edges and vertices, removing the original edges, replacing them with rectangles, removing the original vertices, and replacing them with...

a deltoidal icositetrahedron, which has eighteen order-4 vertices and eight order-3 vertices. This is not to be confused with the dodecagonal trapezohedron...

complete graph K8. The vertices of the Gosset graph can be identified with two copies of the set of edges of K8. Two vertices of the Gosset graph that...

vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of opposite vertices, are the vertices...

an ordered pair G := (V, E) with V a set of vertices or nodes, E a multiset of unordered pairs of vertices, called edges or lines. A multigraph G is an...

Archimedean solid, in which "identical vertices" means merely that the parts of the polyhedron near any two vertices look the same (they have the same shapes...

worst case to store the stack of vertices on the current search path as well as the set of already-visited vertices. Thus, in this setting, the time and...

a quotient graph Q of a graph G is a graph whose vertices are blocks of a partition of the vertices of G and where block B is adjacent to block C if some...

are vertices of a square). Draw the bisectors of the right angles GOA and EOG, making two more diameters HOD and FOB. A,B,C,D,E,F,G,H are the vertices of...

while in other problems any permutation of the vertices is considered to form the same graph, so the vertices are considered identical or unlabeled. In general...

(4-orthoplex), which means its vertices lie in opposite pairs on the 4 axes of a (w, x, y, z) Cartesian coordinate system. The eight vertices are (±1, 0, 0, 0), (0...

formed from a subset of the vertices of the graph and all of the edges, from the original graph, connecting pairs of vertices in that subset. Formally,...

run-time of each phase is O(m + n). If the number of vertices before a phase is n', the number of vertices remaining after a phase is at most n ′ 2 m / n ′...

degrees of pairs of non-adjacent vertices: if every such pair has a sum that at least equals the total number of vertices in the graph, then the graph is...

from the vertices and edges of the hypercube. For instance, the cube graph Q 3 {\displaystyle Q_{3}} is the graph formed by the 8 vertices and 12 edges...

concurrent. If a hexagon has vertices on the circumcircle of an acute triangle at the six points (including three triangle vertices) where the extended altitudes...

bigon, digon, or a 2-gon, is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the...

A convex polygon is the convex hull of its vertices. Thus it is fully defined by the set of its vertices, and one only needs the corners of the polygon...

polytope acts transitively on its vertices, or that the vertices lie within a single symmetry orbit. All vertices of a finite n-dimensional isogonal...

representing a binary relation, whose vertices V are partitioned between left vertices in X and right vertices in Y, and edges in E always connect a left...

clique (/ˈkliːk/ or /ˈklɪk/) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique...

sides (or edges) and two vertices. In Euclidean space, digons are degenerate, collapsing to a line segment between the two vertices. In spherical geometry...

distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing...

them, and adds a new face between each two adjacent faces (moving the vertices inward). Oppositely, similar to expansion, it moves the faces apart outward...

between the subsets of vertices A,B ⊆ V(G). In the equation, the minimum is over all nonempty sets S of at most n⁄2 vertices and ∂S is the edge boundary...

outer boundary of a subset S of the vertices of a graph G is the set of vertices in G that are adjacent to vertices in S, but not in S themselves. The...

for any polytope whose vertices have a bijection with the permutations of some set. The permutohedron of order n has n! vertices, each of which is adjacent...

simple hypergraph as the number of hypergraph vertices that belong to a hyperedge of three or more vertices. He shows that, for any fixed value of L, a...

considering pairs of adjacent vertices (i.e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart...

set of n {\displaystyle n} vertices into r {\displaystyle r} parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong...

of vertices, and the function w assigns a nonnegative real weight to every edge of G. According to the triangle inequality, for every three vertices u...

(it has more than k vertices and it remains connected after removing fewer than k vertices) if and only if every pair of vertices has at least k internally...

has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share...

which one or more components have an odd number of vertices (even if the total number of vertices is even). Let us call such components odd components...

undirected graph, there is an equivalence relation on the vertices according to which two vertices are related to each other whenever there are two edge-disjoint...

s} vertices of U {\displaystyle U} and t {\displaystyle t} vertices of V {\displaystyle V} that are all connected to each other, then these vertices induce...

simple Lie group SO(8) form the vertices of a 24-cell. The vertices can be seen in 3 hyperplanes, with the 6 vertices of an octahedron cell on each of...

the shortest path between two vertices, until the estimate is optimal. Consider a graph G {\displaystyle G} with vertices V {\displaystyle V} numbered...

undirected graph. That is, it lists all subsets of vertices with the two properties that each pair of vertices in one of the listed subsets is connected by...

toppling of one vertex can render other vertices unstable. Thus, repeat the toppling procedure until all vertices of z i {\displaystyle z_{i}} eventually...

regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges. There are...

each other. The order of a bramble is the minimum number of vertices needed in a set of vertices that hits all of the subgraphs in the family. The set of...

equivalence classes of this relation partition the vertices of the graph into disjoint sets, subsets of vertices that are all reachable from each other, with...

choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado...

meet correspond to the vertices of the octahedron, while the eight vertices where three rhombi meet correspond to the vertices of the cube. The skeleton...

zeroth, the 10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex...

polygon except its vertices are not all coplanar. While a polygon is ordinarily defined as a plane figure, the edges and vertices of a skew polygon form...

calculates a normal based on the order of the triangle vertices using the "right-hand rule", i.e. the vertices are listed in counter-clock-wise order from outside...

equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron. The truncated...

corresponds to the fact that its vertices are the anticomplements of the vertices A, B, and C of the reference triangle. The vertices of the medial triangle are...

sharing the same 27 vertices. The 216 edges in 221 can be seen as the 72 3{} edges represented as 3 simple edges. Its 27 vertices can be given coordinates...

a graph, and let U be an independent set of vertices, that is, U is a subset of V in which no two vertices are connected by an edge. Let NG(U) denote the...

semi-symmetric graph after the Gray graph on 54 vertices and the Iofinova-Ivanov graph on 110 vertices. The characteristic polynomial of the Ljubljana...

vertices of v {\displaystyle v} , it can be colored in G {\displaystyle G} with a color not used by the neighbors. So now look at those five vertices...

{\displaystyle F} from the vertices of T 1 {\displaystyle T_{1}} to the vertices of T 2 {\displaystyle T_{2}} such that: For all vertices v {\displaystyle v}...

there do not exist two vertices the removal of which would disconnect the rest of the graph. A graph is bipartite if its vertices can be colored with two...

the plane in general position has a subset of four points that form the vertices of a convex quadrilateral. This was one of the original results that led...

cases of four and five vertices, the lists below are by no means exhaustive of all possible polyhedra with the given number of vertices, but rather just include...

(vertical-axis vertices, black in first figure); four degree-4 vertices surrounded by three acute and one obtuse face angles (horizontal-axial-plane vertices, blue...

the articles describing various individual (finite) graphs. The columns 'vertices', 'edges', 'radius', 'diameter', 'girth', 'P' (whether the graph is planar)...

edges are the same as the vertices and edges of the regular octahedron, which has six vertices and twelve edges. Six vertices of the octahedral graph can...

E) with v vertices and degree k such that for some given integers λ , μ ≥ 0 {\displaystyle \lambda ,\mu \geq 0} every two adjacent vertices have λ common...

bipyramids as well; the bipyramids vertices correspond to the faces of the prism, and the edges between pairs of vertices of one correspond to the edges between...

When the number of vertices in the graph is known ahead of time, and additional data structures are used to determine which vertices have already been...

any set T of evenly many vertices, and must produce as output a minimum-weight edge set in the graph whose odd-degree vertices are precisely those of T...

plane, sharing one of their acute-angled vertices, in such a way that the remaining two acute-angled vertices are a unit distance apart from each other...

the object's vertices given some formal representation of the object. A classical example is the problem of enumeration of the vertices of a convex polytope...

circle, this is the case when at least one of its vertices is an ideal point or when all of its vertices lie on a horocycle or on a one-sided hypercycle...

{\displaystyle W} of all vertices satisfies | W | / | V | = 8 / 8 = 1 > 1 − ε {\displaystyle |W|/|V|=8/8=1>1-\varepsilon } . Since all 8 vertices have isomorphic...

where every adjacent pair of vertices has the same number l of neighbors in common, and every non-adjacent pair of vertices has the same number n of neighbors...

ligands or interstitial atoms, rather than vertices. Larger and more electropositive atoms tend to occupy vertices of high connectivity and smaller more electronegative...

vertices is between the other two in the path. If the subtree formed by the union of the three paths is not a path, the median of the three vertices is...

angles at the three vertices. The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices using the side lengths...

between opposite pairs of vertices in a 4-dimensional hypercube graph. (In an n-dimensional hypercube, a pair of vertices are opposite if the shortest...

triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector...

compute the total number of vertices and edges. It will have g/r vertices, and g/p edges. When p=r, the number of vertices and edges are equal. This condition...

graph is the graph of vertices and edges of the rhombicosidodecahedron, one of the Archimedean solids. It has 60 vertices and 120 edges, and is a quartic...

set of vertices and a set of unordered pairs of vertices, which are called edges. By convention, in algorithm analysis, the number of vertices in the...

the five degree-four vertices of the Grötzsch graph by a set of three vertices, replacing each of the five degree-three vertices of the Grötzsch graph...

components of a graph. A graph is composed of vertices (nodes) and edges. Each edge is connected to two vertices, and each vertex may be connected to zero...

must be known with respect to the surface vertices to describe the system. In anatomy, the surface vertices of the eye's lens are called the anterior...

vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, Gn has 2n+1 vertices and 3n edges. Gear graphs are examples...

a regular icosahedron by cutting off all of its vertices, known as truncation. Each of the 12 vertices at the one-third mark of each edge creates 12 pentagonal...

7-simplex, or alternately can be seen as an expansion operation. Its 56 vertices represent the root vectors of the simple Lie group A7. Expanded 7-simplex...

vertices out to different radii from the body center. The radii are chosen so that the resulting shape has planar kite faces each such that vertices go...

6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are...

cardinality of a subset S of vertices such that all other vertices are uniquely determined by their distances to the vertices in S. Finding the metric dimension...

consists of all vertices at distance one from v {\displaystyle v} , and the second neighborhood of v {\displaystyle v} consists of all vertices at distance...

theory, a minimum cut or min-cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets) that is minimal in some metric. In...

Red and Blue players take turns coloring the vertices of a graph, with the constraint that two vertices that are connected by an edge may not be colored...

quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and...

Petersen graph G(8,3): that is, it is formed by the vertices of an octagon, connected to the vertices of an eight-point star in which each point of the...

chessboard. Its vertices represent the squares of the chessboard, and may be given coordinates (x, y), where 1 ≤ x ≤ n and 1 ≤ y ≤ m. Two vertices with coordinates...

triangle, 4p square and 4 regular p-gon faces. It has 10p edges, and 4p vertices. There is an infinite series of convex uniform antiprismatic prisms, starting...

with distinct vertices, with each triangle colored with one of three colors, it must be possible to regroup the nine triangle vertices into three "rainbow"...

Tetrahedral graph – 4 vertices, 6 edges Octahedral graph – 6 vertices, 12 edges Cubical graph – 8 vertices, 12 edges Icosahedral graph – 12 vertices, 30 edges Dodecahedral...

of an n {\displaystyle n} -element set; two vertices are adjacent when the intersection of the two vertices (subsets) contains ( k − 1 ) {\displaystyle...

all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched...

The frontier contains all vertices that have the same distance (also called "level") from the source vertex, these vertices need to be explored in BFS...

position (i, j) the distance between vertices vi and vj. The distance is the length of a shortest path connecting the vertices. Unless lengths of edges are explicitly...

graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices have exactly...

sequence of vertices in G such that each two adjacent vertices in the sequence are connected by an edge in G, and each two nonadjacent vertices in the sequence...

of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals n. For a regular simple...

pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries...

following figures show a graph (left, with blue vertices) and its line graph (right, with green vertices). Each vertex of the line graph is shown labeled...

to A0, so Ai = Si(A0). The set of vertices Ai with i any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments...

most 9 vertices satisfying the condition that all edges have positive imbalance. This was further improved to graphs with at most 12 vertices. Several...

whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the...

these three vectors. These 27 vectors correspond to the vertices of the Schläfli graph; two vertices are adjacent if and only if the corresponding two vectors...

in connecting specified "query vertices" in a network. Given a connected, undirected graph and a set of query vertices in a graph, the minimum Wiener...

discovered by S. S. Shrikhande in 1959. It is a strongly regular graph with 16 vertices and 48 edges, with each vertex having degree 6. Every pair of nodes has...

), defined by n + 1 vertices, has a face for each subset of the vertices, from the empty set up through the set of all vertices. In particular, there...

algorithm that repeatedly removes minimum-degree vertices. The connected components that are left after all vertices of degree less than k {\displaystyle k} have...

icosahedron. The icosahedral graph has twelve vertices, the same number of vertices as a regular icosahedron. These vertices are connected by five edges from each...

vice versa. A Möbius transformation can take the vertices of a square (but not its edges) to the vertices of a harmonic quadrilateral. The wallpaper groups...

(v,e,f) (vertices, edges, and faces), will have a new f-vector (1+v, v+e, e+f, f+1). For example, a cube has f-vector (8,12,6), 8 vertices, 12 edges...

of A B C {\displaystyle ABC} hold. Normalizing coordinates at vertices, the three vertices A, B, and C have coordinates: A = 1 : 0 : 0 B = 0 : 1 : 0 C =...

distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing...

graph (vertex 1), and then adding black vertices as isolated vertices and red vertices as dominating vertices, in the order in which they are numbered...

the set of all vertices of minimum eccentricity, that is, the set of all vertices u where the greatest distance d(u,v) to other vertices v is minimal....

unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex...

style of drawing that places the vertices of a graph on a circle, often evenly spaced so that they form the vertices of a regular polygon. Circular layouts...

pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries...

metric optimized here is the travel time. Intersections are represented by vertices, the road sections connecting them by edges. The edge weights represent...

{\displaystyle p_{1}} vertices and q 1 {\displaystyle q_{1}} edges, and G 2 {\displaystyle G_{2}} has p 2 {\displaystyle p_{2}} vertices and q 2 {\displaystyle...

"enantiomorphs") of each other. It has 92 vertices that span 60 pentagonal faces. It is the Catalan solid with the most vertices. Among the Catalan and Archimedean...

n {\displaystyle n} vertices into r {\displaystyle r} subsets, with sizes as equal as possible, and then connecting two vertices by an edge if and only...

four vertices lie on the major axis and two lie on the minor axis. For a circle, which has constant curvature, every point is a vertex. Vertices are points...

graph contains at least 20 vertices. Examples of 4-regular matchstick graphs are currently known for all number of vertices ≥ 52 except for 53, 55, 56...

Robertson graph or (4,5)-cage, is a 4-regular undirected graph with 19 vertices and 38 edges named after Neil Robertson. The Robertson graph is the unique...

complete graphs. Switching a set of vertices in a (simple) graph means reversing the adjacencies of each pair of vertices, one in the set and the other not...

unbounded chromatic number. Every complete graph on n vertices, and by extension every graph on n vertices, has an incidence poset with order dimension O(log log n)...

edges E and vertices V can be expressed as a function of the number of edges, denoted | E | {\displaystyle |E|} , and the number of vertices, denoted |...

graphs on more than three vertices, since these have strictly more edges ( ( n 2 ) {\displaystyle {\binom {n}{2}}} ) than vertices ( n {\displaystyle n} )...

k {\displaystyle k} -fan. Any two vertices having exactly one neighbor in common is equivalent to any two vertices being connected by exactly one path...

that matters is which pairs of vertices are connected by edges. In the concrete, however, the arrangement of these vertices and edges within a drawing affects...

planar graph; it has as many edges as G, as many vertices as G has faces and as many faces as G has vertices. The term "dual" is justified by the fact that...

t(G) = 1. When G is the cycle graph Cn with n vertices, then t(G) = n. For a complete graph with n vertices, Cayley's formula gives the number of spanning...

Given an undirected graph with non-negative edge weights and a subset of vertices, usually referred to as terminals, the Steiner tree problem in graphs requires...

Directed graphs require that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident...

hosohedron, {2,1} has two antipodal vertices at the poles, one 360° lune face, and one edge (meridian) between the two vertices. Look up monogon in Wiktionary...

set of vertices in a graph, no two of which are adjacent. That is, it is a set S {\displaystyle S} of vertices such that for every two vertices in S {\displaystyle...

zeroth being the 10-cube itself. Vertices of the rectified 10-cube are located at the edge-centers of the 10-cube. Vertices of the birectified 10-cube are...

set of vertices that includes at least one endpoint of every edge, and a vertex cover is minimum if no other vertex cover has fewer vertices. A matching...

connect the tree to vertices not yet in the tree, find the minimum-weight edge, and transfer it to the tree. Repeat step 2 (until all vertices are in the tree)...

of vertices in which deleting one vertex in every possible way results in a graph with a perfect matching, a way of grouping the remaining vertices into...

to be vertices. The four-vertex theorem states that a smooth closed curve always has at least four vertices. An ellipse has exactly four vertices: two...

graph if there exists a partial order ≤ on the vertices of G with the property that for any vertices u and v of G, uv is an edge of G if and only if...

any graph without isolated vertices, the sum of the matching number and the edge covering number equals the number of vertices. If there is a perfect matching...

do operations on the vertices at the corners of triangles. With individual triangles, the system has to operate on three vertices for every triangle. In...

called the end-vertices (or endpoints) of the edge e {\displaystyle e} , and the edge e {\displaystyle e} is said to be incident with vertices u {\displaystyle...

a given even number of vertices grows at least exponentially in the number of vertices. (Because they have odd-degree vertices, all snarks must have an...

Cubic graph with 10 vertices and 15 edges Ptolemy's theorem – Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle Seal...

polygon are truncated down to vertices at the center of each original edge. New edges are formed between these new vertices. This construction is not reversible...

two vertices s and t, there exist paths from s to t of every possible length from the distance d(s,t) up to n − 1, where n is the number of vertices in...

= k for all vertices v, f-choosability corresponds to k-choosability. Consider the complete bipartite graph G = K2,4, having six vertices A, B, W, X,...

visited more than once), trails (vertices can be visited multiple times, no edge is traversed more than once), or walks (vertices and edges can be visited/traversed...

of the three vertices of the triangle to the point is the smallest possible or, equivalently, the geometric median of the three vertices. It is so named...

of the vertices into subsets that have the same distance from a given root vertex. Given a connected graph G = (V, E) with V the set of vertices and E...

A_{2},\dots ,A_{n})} called its vertices. The curve itself consists of the line segments connecting the consecutive vertices. A simple polygonal chain is...

rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 10-orthoplex...

correspond to permuting the six vertices of the K6 graph; these permutations act on the Tutte–Coxeter graph by permuting the vertices on each side of its bipartition...

by removing a small number of vertices. Specifically, the removal of ⁠ O ( n ) {\displaystyle O({\sqrt {n}})} ⁠ vertices from an n-vertex graph (where...

(alternatively KGn,k) is the graph whose vertices correspond to the k-element subsets of a set of n elements, and where two vertices are adjacent if and only if the...

removal of any d − 1 vertices leaves a connected subgraph. For instance, for a three-dimensional polyhedron, even if two of its vertices (together with their...

tiles. An isotoxal polygon has two vertices and one edge. There are isotoxal decagram forms, which alternates vertices at two radii. Each form has a freedom...

8 vertices. It has an unexpected property that every cell is in contact with every other cell on two faces, and every cell contains all the vertices, which...

is no augmenting path (a path that starts and ends on free (unmatched) vertices, and alternates between edges in and not in the matching) with M. It was...

runcinations. Small cellated octaexon (acronym: sco) (Jonathan Bowers) The vertices of the stericated 7-simplex can be most simply positioned in 8-space as...

zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex. Vertices of the birectified 7-simplex...

where V is a set of vertices and E is a set of subsets of V called hyperedges. Each hyperedge may contain one or more vertices. A vertex-cover (aka hitting...

The vertices of the first stellation of the rhombic dodecahedron include the 12 vertices of the cuboctahedron, together with eight additional vertices (the...

could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph (V1, V2, E) such that for every two vertices v1 ∈ V1 and...

strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair...

on 4 vertices. Denote the central vertex of each star Ai and the outer vertices Bi, Ci and Di. This results in a disconnected graph on 4n vertices with...

discovered earlier by Ákos Császár (1949); it has seven vertices, 21 edges connecting every pair of vertices, and 14 triangular faces. Like the Szilassi polyhedron...

0 and 1 to the vertices of G with the property that the number of vertices labeled 0 is as close as possible to the number of vertices labeled 1: they...

There are four colors, and seven vertices of each color. Each red, green or blue vertex is connected with two vertices of the same color (thin edges forming...

regular heptagon (where there are seven vertices and edges), and the total number of diagonals between all its vertices. There are fourteen polygons that can...

the vertices of an n-dimensional hypercube, and the Hamming distance of the strings is equivalent to the Manhattan distance between the vertices. The...

complement of a Rook's graph. Its vertices can be placed in an n-by-n grid, so that each vertex is adjacent to the vertices that are not in the same row or...

colors to the vertices of an undirected graph, in such a way that No two adjacent vertices have the same color, and The numbers of vertices in any two color...

graph is a set of vertices of the graph that has a nonempty intersection with every odd cycle in the graph. Removing the vertices of an odd cycle transversal...

complement or inverse of a graph G is a graph H on the same vertices such that two distinct vertices are adjacent (connected) in H if and only if they are not...

labels to vertices, edges and faces of a graph. Defective coloring is a variant of proper vertex coloring. In a proper vertex coloring, the vertices are coloured...

illustration. Notice that it touches four of the cube's eight vertices, and those four vertices are linked by a 3-edge path that makes two right-angled turns...

refinement scheme. Start with a mesh of an arbitrary polyhedron. All the vertices in this mesh shall be called original points. For each face, add a face...

visual magnitude of 0.77 and is one of the vertices of the Summer Triangle asterism; the other two vertices are marked by Deneb and Vega. It is located...

component containing a positive fraction of the vertices. No other component will contain more than O(log(n)) vertices. If p < ( 1 − ε ) ln ⁡ n n {\displaystyle...

connecting two vertices that are not yet connected. Each new diagonal adds one edge and one face and does not change the number of vertices, so it does not...

(plural: vertices) or a corner. Edges and vertices are more formal, but may be ambiguous in contexts that also involve the edges and vertices of a graph;...

circles centered at its vertices. Two more circles, its Soddy circles, are tangent to the three circles centered at the vertices; their centers are called...

graph theory, a part of mathematics, a k-partite graph is a graph whose vertices are (or can be) partitioned into k different independent sets. Equivalently...

removal of fewer than tk vertices. For instance, a graph is 1-tough if the number of components formed by removing a set of vertices is always at most as...

edges and vertices of the result can be calculated from Euler's polyhedral formula: if G {\displaystyle G} has n {\displaystyle n} vertices, it has exactly...

coordinates are the means of the coordinates of the three vertices. That is, if the three vertices are L = ( x L , y L ) , {\displaystyle L=(x_{L},y_{L})...

triangle. That is, the lines connecting the vertices of the tangential triangle and the corresponding vertices of the reference triangle are concurrent....

{\displaystyle |V|} is the number of vertices in the graph. In each of these repetitions, the number of vertices with correctly calculated distances grows...

edges and vertices) Great complex icosidodecahedron (looks like a great icosahedron and a small stellated dodecahedron sharing edges and vertices) Small...

associated with a group of vertices; for example, in a model of a human being, the bone for the thigh would be associated with the vertices making up the polygons...

polygon.) The vertices of the base can be designated ⁠ V 1 {\displaystyle V_{1}} ⁠ through ⁠ V 2 n , {\displaystyle V_{2n},} ⁠ while the vertices of the top...

from the vertices A, B, C of triangle △ABC to the sides EF, FD, DE of triangle △DEF are concurrent then the perpendiculars from the vertices D, E, F of...

(the complete graph on five vertices) nor of K 3 , 3 {\displaystyle K_{3,3}} (a complete bipartite graph on six vertices, three of which connect to each...

same vertices. The name was given to these graphs by Paul Erdős and András Hajnal. To define the half graph on 2 n {\displaystyle 2n} vertices u 1 ,...

a path on n {\displaystyle n} vertices. The fan graph F n {\displaystyle F_{n}} has n + 1 {\displaystyle n+1} vertices and 2 n − 1 {\displaystyle 2n-1}...

technique that shows how to find a large set of vertices in a dense graph such that every small subset of vertices has many common neighbors. It is a useful...

two used vertices to form a triangle), possibly within a triangle mesh. If N is the number of triangles in the fan, the number of vertices describing...

Johnson solid. The edges and vertices of the triaugmented triangular prism form a maximal planar graph with 9 vertices and 21 edges, called the Fritsch...

reachability between all pairs of vertices can be determined by identifying the connected components of the graph. Any pair of vertices in such a graph can reach...

heptagon into seven triangular "pie slices" with vertices at the center and at the heptagon's vertices, and then halving each triangle using the apothem...

lines, and an individual line has two or more vertices and/or nodes. 2) The length between two vertices is calculated on the surface of a sphere. 3) The...

cyclic n-gon have vertices A1 , ..., An on the unit circle. Then for any point M on the minor arc A1An, the distances from M to the vertices satisfy M A 1...

circumcenters of Delaunay triangles are the vertices of the Voronoi diagram. In the 2D case, the Voronoi vertices are connected via edges, that can be derived...

gives an equation that applies to the number of vertices, edges, and faces of any planar graph. The vertices are just the grid points of the polygon; there...

degrees of rectifications. Vertices of the rectified 9-simplex are located at the edge-centers of the 9-simplex. Vertices of the birectified 9-simplex...

rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex...

the vertices of the graph to reduce the bandwidth of the adjacency matrix. The algorithm produces an ordered n-tuple R {\displaystyle R} of vertices which...

be embedded in the plane such that adjacent vertices are exactly at distance one apart, with no two vertices embedded to the same point and no vertex embedded...

ten vertices at infinity. List of uniform polyhedra Hemi-icosahedron - The ten vertices at infinity correspond directionally to the 10 vertices of this...

their vertex configuration: A skew dodecagon is a skew polygon with 12 vertices and edges but not existing on the same plane. The interior of such a dodecagon...

of vertices. Example. A graph G satisfies property P if it is bipartite with an even number of vertices or perfect with an odd number of vertices. In...

the first), reverse all the signs. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the...

there exists a path between every pair of vertices. The fact that there exists a path between two vertices is the transitive closure of the adjacency...

are 8 pentic forms of the 6-cube. The pentic 6-cube, , has half of the vertices of a pentellated 6-cube, . Stericated 6-demicube Stericated demihexeract...

{\displaystyle v,e,f} are the vertices, edges, and faces of the seed (respectively), and v ′ , e ′ , f ′ {\displaystyle v',e',f'} are the vertices, edges, and faces...

makes the forces cancel at all interior vertices, but this is not necessarily an equilibrium stress at the vertices of the outer polygon. However, when the...

30, is represented in its vertices. The rectified 5-orthoplex is the vertex figure of the D5 lattice, . Its 40 vertices represent the kissing number...

components that contains all vertices and is cycle-free, i.e., there is at most one path between each pair of vertices. Given such a forest F with connected...

header which defines a file in ASCII format. There are 14 vertices (6 faces * 4 vertices - 10 vertices saved due to merging) and 6 faces in total. After the...

the rhombicuboctahedron may also be constructed by cutting all edges and vertices of either cube or a regular octahedron, a process known as cantellation...

and an edge-set E. Each edge is a subset of vertices (in a graph, each edge contains at most two vertices, but in a hypergraph, it may contain more than...

inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle. This...

parts, taking vertices to sides and sides to vertices. From any kite, the inscribed circle is tangent to its four sides at the four vertices of an isosceles...

special cases: Graphs that have finitely many vertices of infinite degree Graphs in which all vertices have infinite degree, by an argument using the...

the sum of charges of vertices is 12, therefore there must exist some vertices with positive charge. We consider which vertices could have positive final...

trapezohedron with n-gonal pyramids truncated from its two polar axis vertices. The vertices exist as 4 n-gons in four parallel planes, with alternating orientation...

n1+o(1) vertices. Sumner's conjecture states that tournaments are universal for polytrees, in the sense that every tournament with 2n − 2 vertices contains...

which have turning numbers greater than 1. The vertices with clockwise turning angles equal the vertices with counterclockwise turning angles. A crossed...

of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 edges, and 24 vertices. Since each of its faces...

by adding diagonals from one vertex to all other non-nearest neighbor vertices. The total number of ways to triangulate a convex n-gon by non-intersecting...

5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are...

in a connected graph in which every vertex has at most Δ neighbors, the vertices can be colored with only Δ colors, except for two cases, complete graphs...

with a large number t of vertices per tree. (For a graph with n vertices and r terminals, they use t = n − r − 1 added vertices per tree.) Then, they ask...

which is also the center of the circumcircle that passes through the three vertices). In an isosceles triangle with exactly two equal sides, these three points...

\mathbf {a} {\bigr \|},} where a, b, and c are vectors to the triangle's vertices from any arbitrary origin point, so that b – a and c – a are the translation...

zeroth, the 5-simplex itself. Vertices of the rectified 5-simplex are located at the edge-centers of the 5-simplex. Vertices of the birectified 5-simplex...

articulation vertices. The property of being 2-connected is equivalent to biconnectivity, except that the complete graph of two vertices is usually not...

but any pair of vertices dominates the whole graph. The product C4 □ C4 is a four-dimensional hypercube graph; it has 16 vertices, and any single vertex...

no degree-two vertices by adding a cycle connecting its leaves. Every Halin graph is 3-vertex-connected: deleting two of its vertices cannot disconnect...

mathematical field of graph theory, the Dyck graph is a 3-regular graph with 32 vertices and 48 edges, named after Walther von Dyck. It is Hamiltonian with 120...

hypercube vertices. The folded cube graph of dimension k (containing 2k − 1 vertices) may be formed by adding edges between opposite pairs of vertices in a...

a graph with 2 vertices and 1 edge connecting them, the pebbling number is 2. No matter how the two pebbles are placed on the vertices of the graph it...

triangle's vertices; its radius is the circumradius. Any point on a perpendicular bisector of one side is equidistant from the two adjacent vertices of the...

obtuse-angled vertices, are all the same length. Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an orthocentric...

cantellations. Small prismated heptapeton (Acronym: spil) (Jonathan Bowers) The vertices of the runcinated 6-simplex can be most simply positioned in 7-space as...

mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. It is called...

formed from another graph by selecting a subset of vertices and adding edges between each pair of vertices in the subset. The planted clique problem is the...

base. this polyhedron has 8 faces, 18 edges, and 12 vertices. A hexagonal prism has twelve vertices, eighteen edges, and eight faces. Every prism has two...

unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex...

symmetry The Levi graph of the Miquel configuration. 6 blue vertices from circles, and 8 red vertices from points. The configuration has maximal symmetry in...

line connecting two vertices that are not on the same face. Space diagonals contrast with face diagonals, which connect vertices on the same face (but...

a knight's graph of an m × n {\displaystyle m\times n} chessboard. Its vertices can be represented as the points of the Euclidean plane whose Cartesian...

isosceles triangle faces are truncations of the original vertices. An n-gonal form has 3n vertices, 6n edges, and 2+3n faces: 2 regular n-gons, n rhombi...

obvious. In a k-regular graph, all vertices have degree k. In any graph G with more than one vertex, two vertices in G must have the same degree, so an...

truncated icosahedron. It is a geodesic polyhedron {3,5+}3,0, with pentavalent vertices separated by an edge-direct distance of 3 steps. Geodesic polyhedra are...

of graph consisting of a complete graph (clique) on m vertices and a path graph on n vertices, connected with a bridge. The special case of the (2n/3...

that each G i {\displaystyle G_{i}} has at most i + k {\displaystyle i+k} vertices and for no i < j {\displaystyle i<j} is G i {\displaystyle G_{i}} homeomorphically...

For each pair of non-opposite vertices, there is an edge joining them. More generally, each set of k + 1 orthogonal vertices corresponds to a distinct k-dimensional...

If this is the case, then the m vertices with the largest degrees form a maximum clique in G, and the remaining vertices constitute an independent set....

dual of a regular polyhedron is a regular map whose vertices and edges correspond to the vertices and edges of the original polyhedron, and whose faces...

blocks as a first step in the analysis process. Basic blocks form the vertices or nodes in a control-flow graph. The code in a basic block has: One entry...

endpoints, all other vertices (if any) in augmenting path must be non-free vertices. An augmenting path could consist of only two vertices (both free) and...

transitively on the graph's vertices. In other words, the graph has an automorphism which is a cyclic permutation of its vertices. The graph has an adjacency...

First, match and label the corresponding vertices of the two figures. Second, draw a vector from one of the vertices of one of the figures to the corresponding...

linear matroids. The special case of finding all feedback vertices in a directed graph (vertices that lie on every directed cycle) can be solved in linear...

regular graph with 27 vertices, is claw-free. It is straightforward to verify that a given graph with n {\displaystyle n} vertices and m {\displaystyle...

ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces σ ⊆ V {\displaystyle...

unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex...

of circulant graph in which the vertices can be arranged in a cycle and each vertex is connected to the other vertices whose positions differ by a number...

distance, independent sets of vertices in path graphs, or via distributive lattices. Like the hypercube graph, the vertices of the Fibonacci cube of order...

fullerenes with a given even number of vertices grows quickly in the number of vertices; 26 is the largest number of vertices for which the fullerene structure...

a closed polygon connects each pair of vertices by an edge whenever the line segment connecting the vertices lies entirely in the polygon. It is not...

each of its vertices represents a color: white and black are the top and bottom axial vertices, respectively, and the rest of the vertices are red, blue...

always terminates because the number of vertices in the polytope is finite; moreover since we jump between vertices always in the same direction (that of...

set contains vertices (known as "0-simplices" in this context) and arrows ("1-simplices") between some of these vertices. Two vertices may be connected...

given a partial 9-coloring. The Sudoku graph has 81 vertices, one vertex for each cell. The vertices are labeled with ordered pairs (x, y), where x and...

Bowers) Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20)...

Normalization is aimed to make the influence of such vertices more equal to that of other vertices, by dividing the entries of the Laplacian matrix by...

the set of all vertices of G {\displaystyle G} . First, the vertices of G {\displaystyle G} are divided into essential vertices (vertices which are covered...

removing the vertices of P {\displaystyle P} from G {\displaystyle G} is connected back to P {\displaystyle P} at a limited number of vertices. The precise...

Δ and Γ, a simplicial map is a function  f  that maps the vertices of Δ to the vertices of Γ and that has the property that for any face X of Δ, the...

7-simplex and 2,160 7-orthoplex facets, and 240 vertices. Its vertex figure is the 321 polytope. As its vertices represent the root vectors of the simple Lie...

trapezohedron also exists as a spherical tiling, with 2 vertices on the poles, and alternating vertices equally spaced above and below the equator. Alsina...

has edges between non-adjacent vertices of G. Like clique covers, graph colorings are partitions of the set of vertices, but into subsets with no adjacencies...

In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract...

efficiently determine whether two vertices are part of the same tree. function Kruskal(Graph G) is F:= ∅ for each v in G.Vertices do MAKE-SET(v) for each {u...

elements (points, or vertices) is 4. This matches the 2nd row of the table (1, 4, 4). A cube has 1 cube, 6 faces, 12 edges, and 8 vertices, which corresponds...

to the vertices of the triangulation such that Each of the three vertices A, B, and C of the initial triangle has a distinct color The vertices that lie...

creating new vertices and new faces. The positions of the new vertices in the mesh are computed based on the positions of nearby old vertices, edges, and/or...

iteratively removing vertices from the tree until only two vertices remain. Specifically, consider a labeled tree T with vertices {1, 2, ..., n}. At step...

that the distance between any two vertices in the partial cube is the same as the distance between those vertices in the hypercube. Equivalently, a partial...

where the first sum ranges over the vertices in the interior of M, the second sum is over the boundary vertices, and χ(M) is the Euler characteristic...

may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of vertices, a tiling is known as k-uniform or k-isogonal;...

screen space than it has vertices, interpolating colour values from samples of expensive lighting calculations at vertices is less processor intensive...

that it has 243 vertices, 22 edges per vertex (for a total of 2673 edges), exactly one shared neighbor per pair of adjacent vertices, and exactly two...

pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries...

the same vertex configuration at all vertices but is not vertex-transitive: it is not true that for any two vertices, there exists a symmetry of the polyhedron...

sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer...

Sternberg's triangle of love. The three components, pictorially labeled on the vertices of a triangle, interact with each other and with the actions they produce...

mathematics, the modular graphs are undirected graphs in which every three vertices x, y, and z have at least one median vertex m(x, y, z) that belongs to...

polyhedra are defined as collections of polygons that meet at their edges and vertices, forming a manifold as they do. That is, each edge should be shared by...

pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries...

simplex the point lies in. Then the resulting simplex is composed of the vertices corresponding to an ordered edge traversal from (0, 0, ..., 0) to (1, 1...

spherical surface, such that each lune shares the same two polar opposite vertices. A regular n-gonal hosohedron has Schläfli symbol {2,n}, with each spherical...

dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the...

of its vertices on the unbounded (outside) face of the graph. A 2-outerplanar graph is a planar graph with the property that, when the vertices on the...

caterpillar, there exists a cyclic sequence of all the vertices in which each adjacent pair of vertices in the sequence is at distance one or two from each...

decomposition is a partition of the vertices of a bipartite graph into subsets, with the property that two adjacent vertices belong to the same subset if and...

{\displaystyle d} vertices at distance one from v {\displaystyle v} , and at most d ( d − 1 ) i − 1 {\displaystyle d(d-1)^{i-1}} vertices at distance i {\displaystyle...

only by random tournaments: in a Paley digraph, every small subset of vertices is dominated by some other vertex. Let q be a prime power such that q ≡...

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...

In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube...

G = ( S , T ; E ) {\displaystyle G=(S,T;E)} with n worker vertices (S) and n job vertices (T), and the edges (E) each have a cost c ( i , j ) {\displaystyle...

billboard; corner vertices are automatically generated. Typically used for particle systems Polygon mesh A 3D model consisting of vertices connected by polygon...

graph with 100 vertices and 1100 edges. It is the unique strongly regular graph srg(100,22,0,6), where no neighboring pair of vertices share a common...

{\displaystyle G} having n {\displaystyle n} vertices requires that, if the degrees of the n {\displaystyle n} vertices in increasing order as d 1 ≤ d 2 ≤ . ...

hyperbola. The line through the foci is called the major axis. It contains the vertices V 1 , V 2 {\displaystyle V_{1},V_{2}} , which have distance a {\displaystyle...

Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron. Others share the same vertices and edges as other polyhedron. The...

polygon, a general polygon that can be circumscribed by a circle. The vertices of this polygon are concyclic points. All triangles are cyclic polygons...

the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes...

structure as static and draw a link between vertices according to a particular property of the two vertices involved. Once specified the statistical distribution...

two vertices in a directed graph, which runs in O ( ( log ⁡ n ) 2 ) {\displaystyle O\left((\log n)^{2}\right)} space for n {\displaystyle n} vertices. The...

Take a polygon with n vertices, n edges. Its prism has 2n vertices, 3n edges, and 2 + n faces. Take a polyhedron with V vertices, E edges, and F faces...

graph theory, the ladder graph Ln is a planar, undirected graph with 2n vertices and 3n − 2 edges. The ladder graph can be obtained as the Cartesian product...

complete graphs K n {\displaystyle K_{n}} whose number n {\displaystyle n} of vertices is even, Alspach conjectured that it is always possible to decompose the...

_{\gamma }^{n}} is the graph on the 2n vertices of an n-dimensional unit hypercube [0,1]n in which two vertices are adjacent when their Hamming distance...

is a bipartite graph, the vertices of one part representing the points and the other representing the lines, with two vertices joined if the corresponding...

the sphere, each face being a hemisphere, and vertices on a great circle. It is regular if the vertices are equally spaced. The dual of an n-gonal dihedron...

has 2-fold symmetry along its 6 octahedral vertices. It has the same number of faces, edges, and vertices as the deltoidal icositetrahedron as they are...

the vertices assigned a data value are connected to their neighbors in a 2D topological grid, but the spatial coordinates assigned to the vertices can...

that for almost all placements of its vertices in the plane, there is no way to continuously move its vertices while preserving all edge lengths, other...

of the vertices into sets of size k, it admits a strong coloring. When the order of the graph G is not divisible by k, we add isolated vertices to G just...

{\sqrt {n}}} vertices, and similarly if c ≤ r then choosing a central row will give a separator with at most n {\displaystyle {\sqrt {n}}} vertices. Thus, every...

-dimensional Euclidean space is d + 1 {\displaystyle d+1} , achieved by the vertices of a regular simplex, and the equilateral dimension of a d {\displaystyle...

two-dimensional graph of vertices of the form ( x , y ) {\displaystyle (x,y)} is replaced by a three-dimensional graph of vertices of the form ( x , y ,...

between two vertices already joined by an edge; thus, adding multiple edges preserves planarity. A dipole graph is a graph with two vertices, in which all...

touches five other vertices, called its neighbours. As such, to prove the four colour theorem, it is sufficient to prove that vertices of five or less were...

icosahedron is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices. It can be constructed from the regular icosahedron, with one edge contraction...

equal to the number of vertices. John H. Conway conjectured more generally that every thrackle has at most as many edges as vertices. It is known that the...

a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares. For some classes of tilings on...

the vertices on the major axis have the smallest radius of curvature of any points, R = b 2 a {\textstyle R={b^{2} \over a}} ; and the vertices on the...

separating vertices, splitting the graph at these two vertices into two smaller graphs (with a linked pair of virtual edges having the separating vertices as...

the pair with the smallest cost first, and remove the vertices; then, among the remaining vertices, assign the pair with the smallest cost; and so on. This...

preceding section. The vertices of the associated Dynkin diagram correspond to the roots in Δ. Edges are drawn between vertices as follows, according to...

cut along any plane which separates the given vertex from all the other vertices, but is otherwise arbitrary. This construction determines the combinatorial...

2b. An ellipse has four extreme points: two vertices at the endpoints of the major axis and two co-vertices at the endpoints of the minor axis. Analytically...

features on the map, or the number of vertices in a single line (possibly reduced to the number of salient vertices); such straightforwardness explains...

another directed graph with the same vertices and as few edges as possible, such that for all pairs v, w of vertices, a (directed) path from v to w in D...

and three trapezoid faces. A wedge has five faces, nine edges, and six vertices. A wedge is a polyhedron of a rectangular base, with the faces are two...

its vertices. Polytopes in lower numbers of dimensions have standard names: A polytope comprises elements of different dimensionality such as vertices, edges...

has 5040 vertices, 15120 edges, 16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the...

either internally or externally on the sides of a parallelogram are the vertices of a square. If two lines parallel to sides of a parallelogram are constructed...

s} vertices are adjacent to S {\displaystyle S} , so we have two possible cases. In the first case, S {\displaystyle S} is adjacent to the vertices T 1...

is a set of edges in a directed graph, defined from a partition of the vertices into two subsets, so that each edge that has an endpoint in both subsets...

a Hamiltonian cycle through all its vertices, as well as cycles of almost all lengths up to the number of vertices of the graph. Halin graphs can be recognized...

Möbius–Kantor faces and 27 vertices. It is notable for being the vertex figure of the complex Witting polytope, which shares 240 vertices with the eight-dimensional...

truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex...

graph – 6 vertices, 9 edges Cubical graph – 8 vertices, 12 edges Pentagonal prism graph – 10 vertices, 15 edges Hexagonal prism graph – 12 vertices, 18 edges...

points; two horocycles are concentric when they have the same centre. if all vertices of a triangle are ideal points the triangle is an ideal triangle. Some...

In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016...

(including the orthocenter) coincide at its centroid. Let A, B, C denote the vertices and also the angles of the triangle, and let a = | B C ¯ | , {\displaystyle...

may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of vertices, a tiling is known as k-uniform or k-isogonal;...

{\text{codeg}}(x,y)} of two distinct vertices x {\displaystyle x} and y {\displaystyle y} is the number of edges that contain both vertices. There exists an asymptotic...

follows. A complete graph is an undirected graph in which every pair of vertices form the endpoints of an edge. The rectilinear crossing number of a graph...

Often in operations research, a directed graph is called a network, the vertices are called nodes and the edges are called arcs. A flow must satisfy the...

n-cycle by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is a cubic, circulant graph, so-named because (with the...

vertices of the given graph G be v1, v2, . . . , vn. The Mycielski graph μ(G) contains G itself as a subgraph, together with n+1 additional vertices:...

4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered...

polygon. Since these specular highlights are absent from the polygon's vertices and Gouraud shading interpolates based on the vertex colors, the specular...

farthest distance between any two of its vertices. That is, it is the diameter of a set for the set of vertices of the graph, and for the shortest-path...

Bucket 1 contains vertices of rank 1. Bucket 2 contains vertices of ranks 2 and 3. In general, if the B-th bucket contains vertices with ranks from interval...

with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational...

theory is a part of graph theory. It defines networks as graphs where the vertices or edges possess attributes. Network theory analyses these networks over...

a vertex that is adjacent to a leaf (a vertex of degree one). Support vertices play an important role in the study of domination in graphs, since every...

needed]. The 32 vertices of a 5-cube map into the 22 exterior vertices of the rhombic icosahedron, with the remaining 10 interior vertices forming a pentagonal...

{\displaystyle n+m} vertices, with n {\displaystyle n} vertices on the left representing elements of U {\displaystyle U} , and m {\displaystyle m} vertices on the...

a tree decomposition represents the vertices of a given graph G as subtrees of a tree, in such a way that vertices in G are adjacent only when the corresponding...

and last being the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex...

where A, B, and C denote both the triangle's vertices and the angle measures at those vertices; H is the orthocenter (the intersection of the triangle's...

1 {\displaystyle n-1} of the vertices into a regular polygon, and covers the complete graph in this subset of vertices with ( n − 1 ) / 2 {\displaystyle...

with Schläfli symbol {6/2} in which numbers are placed at each of the six vertices and six intersections, such that the four numbers on each line sum to the...

every undirected graph contains an even number of vertices of odd degree. That is, the number of vertices that have an odd number of incident edges must...

pair of vertices in the top tree ℜ {\displaystyle \Re } can be called as External Boundary Vertices, they can be thought of as Boundary Vertices of the...

at most one (undirected) edge exists between each pair of vertices, and in which no vertices connect to themselves), we have E max = ( N 2 ) = N ( N −...

truncating its vertices of valence four or higher. For instance, truncated trapezohedrons are formed by truncating only the high-degree vertices of a trapezohedron;...

{n\log n}})} vertices, and in some triangle-free graphs every independent set has O ( n log ⁡ n ) {\displaystyle O({\sqrt {n\log n}})} vertices. One way to...

a vertex add up to less than 360° (a defect), while the angles at some vertices of a nonconvex polyhedron may add up to more than 360° (an excess). In...

1280 vertices. These graphs form 89% of the cubic Cayley graphs and 88% of all connected vertex-transitive cubic graphs on the same number of vertices. Unsolved...

each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common...

constructed by orthographically projecting any cube onto a plane through three vertices that are all adjacent to the same vertex. The twelve midpoints to edges...