Formally, a hypergraph is a pair where is a set of elements called nodes or vertices, and is a set of non-empty subsets of called hyperedges or edges.

The partial hypergraph is a hypergraph with some edges removed.

The dual of is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by and whose edges are given by where

The primal graph of a hypergraph is the graph with the same vertices of the hypergraph, and edges between all pairs of vertices contained in the same hyperedge.

When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism.

Consider the hypergraph with edges

The rank of a hypergraph is the maximum cardinality of any of the edges in the hypergraph.

If all edges have the same cardinality k, the hypergraph is said to be uniform or k-uniform, or is called a k-hypergraph.

Similarly, a hypergraph is edge-transitive if all edges are symmetric.

One possible generalization of a hypergraph is to allow edges to point at other edges.

Consider, for example, the generalized hypergraph whose vertex set is and whose edges are and.

For example, consider the generalized hypergraph consisting of two edges and, and zero vertices, so that and.

A hypergraph is a combinatorial structure that, like an undirected graph, has vertices and edges, but in which the edges may be arbitrary sets of vertices rather than having to have exactly two endpoints.

As a special case of this correspondence between bipartite graphs and hypergraphs, any multigraph ( a graph in which there may be two or more edges between the same two vertices ) may be interpreted as a hypergraph in which some hyperedges have equal sets of endpoints, and represented by a bipartite graph that does not have multiple adjacencies and in which the vertices on one side of the bipartition all have degree two.

For example, every polygon is topologically self-dual ( it has the same number of vertices as edges, and these are switched by duality ), but will not in general be geometrically self-dual ( up to rigid motion, for instance ) – regular polygons are geometrically self-dual ( all angles are congruent, as are all edges, so under duality these congruences swap ), but irregular polygons may not be geometrically self-dual.

Players are permitted to play the ball with the flat of the ' face side ' and with the edges of the head and handle of the field hockey stick with the exception that, for reasons of safety, the ball may not be struck ' hard ' with a forehand edge stroke, because of the difficulty of controlling the height and direction of the ball from that stroke.

A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another ; see graph ( mathematics ) for more detailed definitions and for other variations in the types of graph that are commonly considered.

The cut on the right of the neck may have been the result of the body becoming bloated, causing the skin to split ; however, the straight edges to the wound suggest it may have been caused by a sharp instrument such as a knife.

The goal of this problem is to minimize the cost of reaching a target in a weighted graph where some of the edges are unreliable and may have been removed from the graph.

The coversheet is also cut, which may lead to a slight loss of vacuum around the edges of the coversheet, but this loss is usually not significant.

Geometrically two edges meeting at a corner are required to form an angle that is not straight ( 180 °); otherwise, the line segments will be considered parts of a single edge – however mathematically, such corners may sometimes be allowed.

* A random graph, for a graph with V edges, may be parameterized as an NxN matrix, indicating the weight for each edge, or 0 for no edge.

Square, beveled or rounded edges may allow a player to apply different techniques, while tapered rather than straight sides may help improve control and cause less damping.

Both these perspectives are useful: while a tree can be analyzed mathematically as a whole, when actually represented as a data structure it is usually represented and worked with separately by node ( rather than as a list of nodes and an adjacency list of edges between nodes, as one may represent a digraph, for instance ).

There may be several minimum spanning trees of the same weight having a minimum number of edges ; in particular, if all the edge weights of a given graph are the same, then every spanning tree of that graph is minimum.

In other cases intrusive junctions, chilled edges, contact alteration or porphyritic structure may prove that in its original condition a metamorphic gneiss was an igneous rock.

Past Light Cones continue to exist and you may only travel their edges.

The book introduces the concept of spherical triangle ( figures formed of three great circle arcs, which he named " trilaterals ") and proves Menelaus ' theorem on collinearity of points on the edges of a triangle ( which may have been previously known ) and its analog for spherical triangles.

While palette knives are manufactured without sharpened cutting edges, with prolonged use they may become " sharpened " by the action of abrasive pigments such as earth colors.

Flats may or may not be hemmed or otherwise finished at the edges.

FRP rods may also be useful in situations where it is likely that the concrete structure may be compromised in future years, for example the edges of balconies when balustrades are replaced and bathroom floors in multi-story construction where the service life of the floor structure is likely to be many times the service life of the waterproofing building membrane.

A toroidal embedding of K < sub > 3, 3 </ sub > may be obtained by replacing the crossing by a tube, as described above, in which the two holes where the tube connects to the plane are placed along one of the crossing edges on either side of the crossing.

Cracking is limited mostly to drilled discs, which may develop small cracks around edges of holes drilled near the edge of the disc due to the disc's uneven rate of expansion in severe duty environments.

They form the outside edges of the door.

They form the outside edges of the door.

It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a graph ( sometimes called a Schlegel diagram ) on the sphere or on a flat plane, and the corresponding graph formed by the dual of this polyhedron is its dual graph.

The most common geometric arrangement is where some convex polyhedron is in its canonical form, which is to say that the all its edges must be tangent to a certain sphere whose centre coincides with the centre of gravity ( average position ) of the tangent points.

However, over the last hundred years or so, authorities have recognized that the dagger, in its contemporary or mature form, has come to incorporate certain definable characteristics, including a short blade with a sharply-tapered point, a central spine or fuller, and ( usually ) two cutting edges sharpened the full length of the blade, or nearly so.

At this time, slums were also beginning to form in the inner city, and those who could afford to leave, left for the suburbs and outer edges of the city.

For example, the Inuit used chips of the Cape York meteorite to form cutting edges for tools and spear tips.

" Raclette scrapers " are notable for their particular form, being blades or flakes whose edges have been sharply retouched until they are semicircular or even shapeless.

The pieces were then finished by a percussive retouching of the edges ( generally leaving one side with the natural edge of the blank ), giving the piece its definitive polygonal form.

The form of the uppercase letter derives from that of an omicron ( Ο ) broken up at the side ( x16px ), with the edges subsequently turned outwards ( x16px, x16px, x16px ).

The modern lowercase shape goes back to the uncial form x14px, a form that developed during the 3rd century BC in ancient handwriting on papyrus, from a flattened-out form of the letter ( x16px ) that had its edges curved even further upwards.

The paper was supplied in roll form and had perforations along both edges that were engaged by sprockets on the rollers.

The grit wheels at opposite edges of the sheet press against resilient urethane-coated rollers and form tiny indentations in the sheet.

Advanced Covering allows a surface to be fit onto circuits ( collections of edges that form closed loops ) in solid or wire bodies, which is useful in consumer product design.

Cirriform-category clouds form at high tropospheric altitudes along the very leading edges of a frontal or low-pressure weather disturbance and often along the fringes of its other borders.

) This can be visualized as taping the left and right edges of the rectangle to form a tube, then taping the top and bottom edges of the tube to form a torus ( doughnut shape ).

# Discrete discernible " edges " that form an enclosed area completely visible in plan and therefore stratigraphically later than the surrounding surface or

They form colonies by nesting in burrows tunnelled into the side of sandy banks, such as those that have collapsed on the edges of rivers.

Individually cast curved bronze plates square with turned in edges were joined together by rivets through holes formed during casting to form a series of rings.