ArticleslgStudy

mathematics

Planar separator theorem

Planar separator theorem is a mathematics 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 Planar separator theorem rather than just read about it. In short: In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of ⁠ O ( n ) {\displaystyle O({\sqrt {n}})} ⁠ vertices from an n-vertex graph (where the O invokes big O notation) can partition the graph into disjoint subgraphs each of which has at…

Planar separator theorem — main illustration
Planar separator theorem — illustration

Key takeaways

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

Reference excerpt

In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of ⁠ O ( n ) {\displaystyle O({\sqrt {n}})} ⁠ vertices from an n-vertex graph (where the O invokes big O notation) can partition the graph into disjoint subgraphs each of which has at most ⁠ 2 n / 3 {\displaystyle 2n/3} ⁠ vertices. A weaker form of the separator theorem with ⁠ O ( n log 3 / 2 ⁡ n ) {\displaystyle O({\sqrt {n}}\log ^{3/2}n)} ⁠ vertices in the separator instead of ⁠ O ( n ) {\displaystyle O({\sqrt {n}})} ⁠ was originally proven by Ungar (1951), and the form with the tight asymptotic bound on the separator size was first proven by Lipton & Tarjan (1979). Since their work, the separator theorem has been reproven in several different ways, the constant in the ⁠ O ( n ) {\displaystyle O({\sqrt {n}})} ⁠ term of the theorem has been improved, and it has been extended to certain classes of nonplanar graphs. Repeated application of the separator theorem produces a separator hierarchy which may take the form of either a tree decomposition or a branch-decomposition of the graph. Separator hierarchies may be used to devise efficient divide and conquer algorithms for planar graphs, and dynamic programming on these hierarchies can be used to devise exponential time and fixed-parameter tractable algorithms for solving NP-hard optimization problems on these graphs. Separator hierarchies may also be used in nested dissection, an efficient variant of Gaussian elimination for solving sparse systems of linear equations arising from finite element methods. Beyond planar graphs, separator theorems have been applied to other classes of graphs including graphs excluding a fixed minor, nearest neighbor graphs, and finite element meshes. The existence of a separator theorem for a class of graphs can be formalized and quantified by the concepts of treewidth and polynomial expansion.

… excerpt ends here. Continue reading the full article.

Illustrations

Planar separator theorem: A polyhedron formed by replacing each of the faces of an icosahedron by a mesh of 100 triangles, an example of the lower bound construction of Djidjev (1982)
A polyhedron formed by replacing each of the faces of an icosahedron by a mesh of 100 triangles, an example of the lower bound construction of Djidjev (1982)
Planar separator theorem: An intersection graph of disks, with at most k = 5 {\displaystyle k=5} disks covering any point of the plane
An intersection graph of disks, with at most k = 5 {\displaystyle k=5} disks covering any point of the plane

Worked examples

Example 1 — a first encounter with Planar separator theorem

Start with the simplest possible case. Write down what Planar separator theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Planar separator theorem 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 Planar separator theorem 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 Planar separator theorem

In research
Planar separator theorem appears in mathematics 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 Planar separator theorem 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
Planar separator theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Statements about planar graphs, so understanding it makes those chapters shorter.
In everyday life
Look for Planar separator theorem 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Planar separator theorem in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Planar separator theorem means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Planar separator theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Planar separator theorem in simple terms?

In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of ⁠ O ( n ) {\displaystyle O({\sqrt {n}})} ⁠ vertices from an…

Why does Planar separator theorem matter?

Because it connects several mathematics 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 Planar separator theorem?

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 Planar separator theorem.

Tags

  • Statements about planar graphs

Keep exploring