science

Whitney's planarity criterion

Whitney's planarity criterion 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 Whitney's planarity criterion rather than just read about it. In short: In mathematics, Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney. It states that a graph G is planar if and only if its graphic matroid is also cographic (that is, it is the dual matroid of another graphic matroid).

Key takeaways

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

Reference excerpt

In mathematics, Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney. It states that a graph G is planar if and only if its graphic matroid is also cographic (that is, it is the dual matroid of another graphic matroid). In purely graph-theoretic terms, this criterion can be stated as follows: There must be another (dual) graph G′ = (V′,E′) and a bijective correspondence between the edges E′ and the edges E of the original graph G, such that a subset T of E forms a spanning tree of G if and only if the edges corresponding to the complementary subset E − T form a spanning tree of G′.

Algebraic duals An equivalent form of Whitney's criterion is that a graph G is planar if and only if it has a dual graph whose graphic matroid is dual to the graphic matroid of G. A graph whose graphic matroid is dual to the graphic matroid of G is known as an algebraic dual of G. Thus, Whitney's planarity criterion can be expressed succinctly as: a graph is planar if and only if it has an algebraic dual.

Topological duals If a graph is embedded into a topological surface such as the plane, in such a way that every face of the embedding is a topological disk, then the dual graph of the embedding is defined as the graph (or in some cases multigraph) H that has a vertex for every face of the embedding, and an edge for every adjacency between a pair of faces. According to Whitney's criterion, the following conditions are equivalent:

The surface on which the embedding exists is topologically equivalent to the plane, sphere, or punctured plane H is an algebraic dual of G Every simple cycle in G corresponds to a minimal cut in H, and vice versa Every simple cycle in H corresponds to a minimal cut in G, and vice versa Every spanning tree in G corresponds to the complement of a spanning tree in H, and vice versa. It is possible to define dual graphs of graphs embedded on nonplanar surfaces such as the torus, but these duals do not generally have the correspondence between cuts, cycles, and spanning trees required by Whitney's criterion.

References

Illustrations

Worked examples

Example 1 — a first encounter with Whitney's planarity criterion

Start with the simplest possible case. Write down what Whitney's planarity criterion 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 Whitney's planarity criterion 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 Whitney's planarity criterion 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 Whitney's planarity criterion

In research: Whitney's planarity criterion 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 Whitney's planarity criterion 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: Whitney's planarity criterion is common in secondary-school and first-year university syllabi. It links to neighbouring topics Matroid theory, Statements about planar graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Whitney's planarity criterion 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 Whitney's planarity criterion in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Whitney's planarity criterion 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 Whitney's planarity criterion out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Whitney's planarity criterion in simple terms?

Why does Whitney's planarity criterion 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 Whitney's planarity criterion?

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 Whitney's planarity criterion.