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K-outerplanar graph

K-outerplanar graph 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 K-outerplanar graph rather than just read about it. In short: In graph theory, a k-outerplanar graph is a planar graph that has a planar embedding in which the vertices belong to at most k {\displaystyle k} concentric layers. The outerplanarity index of a planar graph is the minimum value of k {\displaystyle k} for which it is k {\displaystyle k} -outerplanar.

K-outerplanar graph — main illustration
K-outerplanar graph — illustration

Key takeaways

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

Reference excerpt

In graph theory, a k-outerplanar graph is a planar graph that has a planar embedding in which the vertices belong to at most k {\displaystyle k} concentric layers. The outerplanarity index of a planar graph is the minimum value of k {\displaystyle k} for which it is k {\displaystyle k} -outerplanar.

Definition An outerplanar graph (or 1-outerplanar graph) has all of its vertices on the unbounded (outside) face of the graph. A 2-outerplanar graph is a planar graph with the property that, when the vertices on the unbounded face are removed, the remaining vertices all lie on the newly formed unbounded face. And so on. More formally, a graph is k {\displaystyle k} -outerplanar if it has a planar embedding such that, for every vertex, there is an alternating sequence of at most k {\displaystyle k} faces and k {\displaystyle k} vertices of the embedding, starting with the unbounded face and ending with the vertex, in which each consecutive face and vertex are incident to each other.

Properties and applications The k {\displaystyle k} -outerplanar graphs have treewidth at most 3 k − 1 {\displaystyle 3k-1} . However, some bounded-treewidth planar graphs such as the nested triangles graph may be k {\displaystyle k} -outerplanar only for very large k {\displaystyle k} , linear in the number of vertices. Baker's technique covers a planar graph with a constant number of k {\displaystyle k} -outerplanar graphs and uses their low treewidth in order to quickly approximate several hard graph optimization problems. In connection with the GNRS conjecture on metric embedding of minor-closed graph families, the k {\displaystyle k} -outerplanar graphs are one of the most general classes of graphs for which the conjecture has been proved. A conjectured converse of Courcelle's theorem, according to which every graph property recognizable on graphs of bounded treewidth by finite state tree automata is definable in the monadic second-order logic of graphs, has been proven for the k {\displaystyle k} -outerplanar graphs.

Recognition The smallest value of k {\displaystyle k} for which a given graph is k {\displaystyle k} -outerplanar (its outerplanarity index) can be computed in quadratic time.

References

Illustrations

K-outerplanar graph: A 3-outerplanar graph, the graph of a rhombic dodecahedron. There are four vertices on the outside face, eight vertices on the second layer (light yellow), and two vertices on the third layer (darker yellow). Because of the symmetries of the graph, no other embedding has fewer layers.
A 3-outerplanar graph, the graph of a rhombic dodecahedron. There are four vertices on the outside face, eight vertices on the second layer (light yellow), and two vertices on the third layer (darker yellow). Because of the symmetries of the graph, no other embedding has fewer layers.

Worked examples

Example 1 — a first encounter with K-outerplanar graph

Start with the simplest possible case. Write down what K-outerplanar graph 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 K-outerplanar graph 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 K-outerplanar graph 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 K-outerplanar graph

In research
K-outerplanar graph 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 K-outerplanar graph 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
K-outerplanar graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Planar graphs, so understanding it makes those chapters shorter.
In everyday life
Look for K-outerplanar graph 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 K-outerplanar graph in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what K-outerplanar graph 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 K-outerplanar graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is K-outerplanar graph in simple terms?

In graph theory, a k-outerplanar graph is a planar graph that has a planar embedding in which the vertices belong to at most k {\displaystyle k} concentric layers. The outerplanarity index of a planar graph is the minimum value of k {\displaystyle k} for which it is k {\displaystyle k} -outerplanar.

Why does K-outerplanar graph 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 K-outerplanar graph?

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 K-outerplanar graph.

Tags

  • Planar graphs

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