mathematics

Rank-width

Rank-width 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 Rank-width rather than just read about it. In short: Rank-width is a graph width parameter used in graph theory and parameterized complexity, and defined using linear algebra. It is defined from hierarchical clusterings of the vertices of a given graph, which can be visualized as ternary trees having the vertices as their leaves.

Key takeaways

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

Reference excerpt

Rank-width is a graph width parameter used in graph theory and parameterized complexity, and defined using linear algebra. It is defined from hierarchical clusterings of the vertices of a given graph, which can be visualized as ternary trees having the vertices as their leaves. Removing any edge from such a tree disconnects it into two subtrees and partitions the vertices into two subsets. The graph edges that cross from one side of the partition to the other can be described by a biadjacency matrix; for the purposes of rank-width, this matrix is defined over the finite field GF(2) rather than using real numbers. The rank-width of a graph is the maximum of the ranks of the biadjacency matrices, for a clustering chosen to minimize this maximum. Rank-width is closely related to clique-width: k ≤ c ≤ 2 k + 1 − 1 {\displaystyle k\leq c\leq 2^{k+1}-1} , where c {\displaystyle c} is the clique-width and k {\displaystyle k} the rank-width. However, clique-width is NP-hard to compute, for graphs of large clique-width, and its parameterized complexity is unknown. In contrast, testing whether the rank-width is at most a constant k {\displaystyle k} takes polynomial time, and even when the rank-width is not constant it can be approximated, with a constant approximation ratio, in polynomial time. For this reason, rank-width can be used as a more easily computed substitute for clique-width. An example of a family of graphs with high rank-width is provided by the square grid graphs. For an n × n {\displaystyle n\times n} grid graph, the rank-width is exactly n − 1 {\displaystyle n-1} . Trees have rank-width at most 1, and the graphs with rank-width at most 1 are precisely distance-hereditary graphs. Graphs of small rank-width are precisely pivot-minors of graphs of small tree-width. A connected graph G with n {\displaystyle n} vertices and m {\displaystyle m} edges has a rank-width of at most m − n + 2 {\displaystyle m-n+2} . A simple proof is to consider a spanning tree and note that trees have rank-width 1. Adding an edge to a graph increases the cut-rank function by at most 1 which increases the rank-width by at most 1, so adding the m − n + 1 {\displaystyle m-n+1} extra edges to the spanning tree increases the rank-width by at most m − n + 1 {\displaystyle m-n+1} .

References

Worked examples

Example 1 — a first encounter with Rank-width

Start with the simplest possible case. Write down what Rank-width 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 Rank-width 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 Rank-width 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 Rank-width

In research: Rank-width 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 Rank-width 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: Rank-width is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph minor theory, Graph theory stubs, Linear algebra, so understanding it makes those chapters shorter.
In everyday life: Look for Rank-width 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 Rank-width in 20 minutes

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

Frequently asked questions

What is Rank-width in simple terms?

Why does Rank-width 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 Rank-width?

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 Rank-width.