In
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, an orientation of an
undirected graph is an assignment of a direction to each edge, turning the initial graph into a
directed graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
Definition
In formal terms, a directed graph is an ordered pai ...
.
Oriented graphs
A directed graph is called an oriented graph if none of its pairs of vertices is linked by two symmetric edges. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of and may be arrows of the graph).
A
tournament
A tournament is a competition involving at least three competitors, all participating in a sport or game. More specifically, the term may be used in either of two overlapping senses:
# One or more competitions held at a single venue and concentr ...
is an orientation of a
complete graph
In 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 ...
. A
polytree
In mathematics, and more specifically in graph theory, a polytree (also called directed tree, oriented tree; . or singly connected network.) is a directed acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its ...
is an orientation of an undirected
tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
.
Sumner's conjecture
Sumner's conjecture (also called Sumner's universal tournament conjecture) states that every orientation of every n-vertex tree is a subgraph of every (2n-2)-vertex tournament.
David Sumner, a graph theorist at the University of South Carolina ...
states that every tournament with vertices contains every polytree with vertices.
The number of
non-isomorphic oriented graphs with vertices (for ) is
: 1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, … .
Tournaments are in one-to-one correspondence with complete directed graphs (graphs in which there is a directed edge in one or both directions between every pair of distinct vertices). A complete directed graph can be converted to an oriented graph by removing every 2-cycle, and conversely an oriented graph can be converted to a complete directed graph by adding a 2-cycle between every pair of vertices that are not endpoints of an edge; these correspondences are
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
. Therefore, the same sequence of numbers also solves the
graph enumeration problem for complete digraphs. There is an explicit but complicated formula for the numbers in this sequence.
Constrained orientations
A
strong orientation is an orientation that results in a
strongly connected graph. The closely related totally cyclic orientations are orientations in which every edge belongs to at least one simple cycle. An orientation of an undirected graph is totally cyclic if and only if it is a strong orientation of every
connected component of .
Robbins' theorem states that a graph has a strong orientation if and only if it is
2-edge-connected; disconnected graphs may have totally cyclic orientations, but only if they have no
bridges.
An
acyclic orientation is an orientation that results in a
directed acyclic graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one v ...
. Every graph has an acyclic orientation; all acyclic orientations may be obtained by placing the vertices into a sequence, and then directing each edge from the earlier of its endpoints in the sequence to the later endpoint. The
Gallai–Hasse–Roy–Vitaver theorem states that a graph has an acyclic orientation in which the longest path has at most vertices if and only if it can be
colored
''Colored'' (or ''coloured'') is a racial descriptor historically used in the United States during the Jim Crow Era to refer to an African American. In many places, it may be considered a slur, though it has taken on a special meaning in South ...
with at most colors. Acyclic orientations and totally cyclic orientations are related to each other by
planar duality. An acyclic orientation with a single source and a single sink is called a
bipolar orientation.
A
transitive orientation is an orientation such that the resulting directed graph is its own
transitive closure. The graphs with transitive orientations are called
comparability graph
In graph theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called transitively orientable graphs, partially orderable graph ...
s; they may be defined from a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...
by making two elements adjacent whenever they are comparable in the partial order. A transitive orientation, if one exists, can be found in linear time. However, testing whether the resulting orientation (or any given orientation) is actually transitive requires more time, as it is equivalent in complexity to
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
.
An
Eulerian orientation of an undirected graph is an orientation in which each vertex has equal in-degree and out-degree. Eulerian orientations of
grid graphs arise in
statistical mechanics in the theory of
ice-type models.
A
Pfaffian orientation has the property that certain even-length cycles in the graph have an odd number of edges oriented in each of the two directions around the cycle. They always exist for
planar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...
s, but not for certain other graphs. They are used in the
FKT algorithm for counting perfect matchings.
See also
*
Connex relation
References
External links
*
*{{mathworld, OrientedGraph, Oriented Graph, mode=cs2
Graph theory objects