science

Orientation (graph theory)

Orientation (graph theory) is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Orientation (graph theory) rather than just read about it. In short: In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph. Oriented graphs A directed graph is called an oriented graph if none of its pairs of vertices is linked by two mutually symmetric edges.

Key takeaways

Orientation (graph theory) belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Orientation (graph theory) to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Orientation (graph theory) from memory before moving on to harder problems.

Reference excerpt

In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph.

Oriented graphs A directed graph is called an oriented graph if none of its pairs of vertices is linked by two mutually symmetric edges. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of (x, y) and (y, x) may be arrows of the graph). A tournament is an orientation of a complete graph. A polytree is an orientation of an undirected tree. Sumner's conjecture states that every tournament with 2n − 2 vertices contains every polytree with n vertices. The number of non-isomorphic oriented graphs with n vertices (for n = 1, 2, 3, …) is

1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, … (sequence A001174 in the OEIS). 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. 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 G is totally cyclic if and only if it is a strong orientation of every connected component of G. 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. 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 k vertices if and only if it can be colored with at most k 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 graphs; they may be defined from a partially ordered set 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. 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 graphs, but not for certain other graphs. They are used in the FKT algorithm for counting perfect matchings.

Illustrations

Worked examples

Example 1 — a first encounter with Orientation (graph theory)

Start with the simplest possible case. Write down what Orientation (graph theory) claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Orientation (graph theory) before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Orientation (graph theory) ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Orientation (graph theory)

In research: Orientation (graph theory) appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Orientation (graph theory) in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Orientation (graph theory) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph theory objects, so understanding it makes those chapters shorter.
In everyday life: Look for Orientation (graph theory) outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Orientation (graph theory) in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Orientation (graph theory) means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Orientation (graph theory) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Orientation (graph theory) in simple terms?

In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph. Oriented graphs A directed graph is called an oriented graph if none of its pairs of vertices is linked by two mutually symmetric edges.

Why does Orientation (graph theory) matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Orientation (graph theory)?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Orientation (graph theory).