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Hamiltonian completion

Hamiltonian completion 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 Hamiltonian completion rather than just read about it. In short: The Hamiltonian completion problem is to find the minimal number of edges to add to a graph to make it Hamiltonian. The problem is clearly NP-hard in the general case (since its solution gives an answer to the NP-complete problem of determining whether a given graph has a Hamiltonian cycle).

Hamiltonian completion — main illustration
Hamiltonian completion — illustration

Key takeaways

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

Reference excerpt

The Hamiltonian completion problem is to find the minimal number of edges to add to a graph to make it Hamiltonian. The problem is clearly NP-hard in the general case (since its solution gives an answer to the NP-complete problem of determining whether a given graph has a Hamiltonian cycle). The associated decision problem of determining whether K edges can be added to a given graph to produce a Hamiltonian graph is NP-complete. Moreover, Hamiltonian completion belongs to the APX complexity class, i.e., it is unlikely that efficient constant ratio approximation algorithms exist for this problem. The problem may be solved in polynomial time for certain classes of graphs, including series–parallel graphs and their subgraphs, which include outerplanar graphs, as well as for a line graph of a tree or a cactus graph. Gamarnik et al. use a linear time algorithm for solving the problem on trees to study the asymptotic number of edges that must be added for sparse random graphs to make them Hamiltonian.

References

Worked examples

Example 1 — a first encounter with Hamiltonian completion

Start with the simplest possible case. Write down what Hamiltonian completion 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 Hamiltonian completion 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 Hamiltonian completion 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 Hamiltonian completion

In research
Hamiltonian completion 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 Hamiltonian completion 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
Hamiltonian completion is common in secondary-school and first-year university syllabi. It links to neighbouring topics Hamiltonian paths and cycles, NP-complete problems, so understanding it makes those chapters shorter.
In everyday life
Look for Hamiltonian completion 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 Hamiltonian completion in 20 minutes

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

Frequently asked questions

What is Hamiltonian completion in simple terms?

The Hamiltonian completion problem is to find the minimal number of edges to add to a graph to make it Hamiltonian. The problem is clearly NP-hard in the general case (since its solution gives an answer to the NP-complete problem of determining whether a given graph has a Hamiltonian cycle).

Why does Hamiltonian completion 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 Hamiltonian completion?

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 Hamiltonian completion.

Tags

  • Hamiltonian paths and cycles
  • NP-complete problems

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