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Tree-depth

Tree-depth 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 Tree-depth rather than just read about it. In short: In graph theory, the tree-depth of a connected undirected graph G {\displaystyle G} is a numerical invariant of G {\displaystyle G} , the minimum height of a Trémaux tree for a supergraph of G {\displaystyle G} . This invariant and its close relatives have gone under many different names in the literature, including vertex ranking number, ordered chromatic number, and minimum elimination tree height; it is also clos…

Tree-depth — main illustration
Tree-depth — illustration

Key takeaways

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

Reference excerpt

In graph theory, the tree-depth of a connected undirected graph G {\displaystyle G} is a numerical invariant of G {\displaystyle G} , the minimum height of a Trémaux tree for a supergraph of G {\displaystyle G} . This invariant and its close relatives have gone under many different names in the literature, including vertex ranking number, ordered chromatic number, and minimum elimination tree height; it is also closely related to the cycle rank of directed graphs and the star height of regular languages. Intuitively, where the treewidth of a graph measures how far it is from being a tree, this parameter measures how far a graph is from being a star.

Definitions The tree-depth of a graph G {\displaystyle G} may be defined as the minimum height of a forest F {\displaystyle F} with the property that every edge of G {\displaystyle G} connects a pair of nodes that have an ancestor-descendant relationship to each other in F {\displaystyle F} . If G {\displaystyle G} is connected, this forest must be a single tree; it need not be a subgraph of G {\displaystyle G} , but if it is, it is a Trémaux tree for G {\displaystyle G} . The set of ancestor-descendant pairs in F {\displaystyle F} forms a trivially perfect graph, and the height of F {\displaystyle F} is the size of the largest clique in this graph. Thus, the tree-depth may alternatively be defined as the size of the largest clique in a trivially perfect supergraph of G {\displaystyle G} , mirroring the definition of treewidth as one less than the size of the largest clique in a chordal supergraph of G {\displaystyle G} . Another definition is the following:

td ⁡ ( G ) = { 1 , if | G | = 1 ; 1 + min v ∈ V td ⁡ ( G − v ) , if G is connected and | G | > 1 ; max i t d ( G i ) , otherwise ; {\displaystyle \operatorname {td} (G)={\begin{cases}1,&{\text{if }}|G|=1;\\1+\min _{v\in V}\operatorname {td} (G-v),&{\text{if }}G{\text{ is connected and }}|G|>1;\\\max _{i}{\rm {td}}(G_{i}),&{\text{otherwise}};\end{cases}}}

where V {\displaystyle V} is the set of vertices of G {\displaystyle G} and the G i {\displaystyle G_{i}} are the connected components of G {\displaystyle G} . This definition mirrors the definition of cycle rank of directed graphs, which uses strong connectivity and strongly connected components in place of undirected connectivity and connected components.

… excerpt ends here. Continue reading the full article.

Illustrations

Tree-depth: The tree-depths of the complete graph K 4 {\displaystyle K_{4}} and the complete bipartite graph K 3 , 3 {\displaystyle K_{3,3}} are both four, while the tree-depth of the path graph P 7 {\displaystyle P_{7}} is three.
The tree-depths of the complete graph K 4 {\displaystyle K_{4}} and the complete bipartite graph K 3 , 3 {\displaystyle K_{3,3}} are both four, while the tree-depth of the path graph P 7 {\displaystyle P_{7}} is three.

Worked examples

Example 1 — a first encounter with Tree-depth

Start with the simplest possible case. Write down what Tree-depth 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 Tree-depth 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 Tree-depth 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 Tree-depth

In research
Tree-depth 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 Tree-depth 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
Tree-depth is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph coloring, Graph invariants, Graph minor theory, so understanding it makes those chapters shorter.
In everyday life
Look for Tree-depth 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 Tree-depth in 20 minutes

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

Frequently asked questions

What is Tree-depth in simple terms?

In graph theory, the tree-depth of a connected undirected graph G {\displaystyle G} is a numerical invariant of G {\displaystyle G} , the minimum height of a Trémaux tree for a supergraph of G {\displaystyle G} . This invariant and its close relatives have gone under many different names in the lit…

Why does Tree-depth 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 Tree-depth?

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 Tree-depth.

Tags

  • Graph coloring
  • Graph invariants
  • Graph minor theory
  • NP-complete problems

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