science

Tolerance graph

Tolerance 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 Tolerance graph rather than just read about it. In short: In graph theory, a tolerance graph is an undirected graph in which every vertex can be represented by a closed interval and a real number called its tolerance, in such a way that two vertices are adjacent in the graph whenever their intervals overlap in a length that is at least the minimum of their two tolerances. This class of graphs was introduced in 1982 by Martin Charles Golumbic and Clyde Monma, who used them…

Key takeaways

Tolerance 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 Tolerance graph to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Tolerance graph from memory before moving on to harder problems.

Reference excerpt

In graph theory, a tolerance graph is an undirected graph in which every vertex can be represented by a closed interval and a real number called its tolerance, in such a way that two vertices are adjacent in the graph whenever their intervals overlap in a length that is at least the minimum of their two tolerances. This class of graphs was introduced in 1982 by Martin Charles Golumbic and Clyde Monma, who used them to model scheduling problems in which the tasks to be modeled can share resources for limited amounts of time. Every interval graph is a tolerance graph. The complement graph of every tolerance graph is a perfectly orderable graph, from which it follows that the tolerance graphs themselves are perfect graphs. It is NP-complete to determine whether a given graph is a tolerance graph. However, because tolerance graphs are perfect graphs, many algorithmic problems that are hard on other classes of graphs, including graph coloring and the clique problem, can be solved in polynomial time on tolerance graphs.

References

Worked examples

Example 1 — a first encounter with Tolerance graph

Start with the simplest possible case. Write down what Tolerance 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 Tolerance 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 Tolerance 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 Tolerance graph

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

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

Frequently asked questions

What is Tolerance graph in simple terms?

Why does Tolerance 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 Tolerance 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 Tolerance graph.