science

Shallow minor

Shallow minor 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 Shallow minor rather than just read about it. In short: In graph theory, a shallow minor or limited-depth minor is a restricted form of a graph minor in which the subgraphs that are contracted to form the minor have small diameter. Shallow minors were introduced by Plotkin, Rao & Smith (1994), who attributed their invention to Charles E.

Key takeaways

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

Reference excerpt

In graph theory, a shallow minor or limited-depth minor is a restricted form of a graph minor in which the subgraphs that are contracted to form the minor have small diameter. Shallow minors were introduced by Plotkin, Rao & Smith (1994), who attributed their invention to Charles E. Leiserson and Sivan Toledo.

Definition

One way of defining a minor of an undirected graph G is by specifying a subgraph H of G, and a collection of disjoint subsets Si of the vertices of G, each of which forms a connected induced subgraph Hi of H. The minor has a vertex vi for each subset Si, and an edge vivj whenever there exists an edge from Si to Sj that belongs to H. In this formulation, a d-shallow minor (alternatively called a shallow minor of depth d) is a minor that can be defined in such a way that each of the subgraphs Hi has radius at most d, meaning that it contains a central vertex ci that is within distance d of all the other vertices of Hi. Note that this distance is measured by hop count in Hi, and because of that it may be larger than the distance in G.

Special cases Shallow minors of depth zero are the same thing as subgraphs of the given graph. For sufficiently large values of d (including all values at least as large as the number of vertices), the d-shallow minors of a given graph coincide with all of its minors.

Classification of graph families Nešetřil & Ossona de Mendez (2012) use shallow minors to partition the families of finite graphs into two types. They say that a graph family F is somewhere dense if there exists a finite value of d for which the d-shallow minors of graphs in F consist of every finite graph. Otherwise, they say that a graph family is nowhere dense. This terminology is justified by the fact that, if F is a nowhere dense class of graphs, then (for every ε > 0) the n-vertex graphs in F have O(n1 + ε) edges; thus, the nowhere dense graphs are sparse graphs. A more restrictive type of graph family, described similarly, are the graph families of bounded expansion. These are graph families for which there exists a function f such that the ratio of edges to vertices in every d-shallow minor is at most f(d). If this function exists and is bounded by a polynomial, the graph family is said to have polynomial expansion.

Separator theorems As Plotkin, Rao & Smith (1994) showed, graphs with excluded shallow minors can be partitioned analogously to the planar separator theorem for planar graphs. In particular, if the complete graph Kh is not a d-shallow minor of an n-vertex graph G, then there exists a subset S of G, with size O(dh2 log n + n/d), such that each connected component of G\S has at most 2n/3 vertices. The result is constructive: there exists a polynomial time algorithm that either finds such a separator, or a d-shallow Kh minor. As a consequence they showed that every minor-closed graph family obeys a separator theorem almost as strong as the one for planar graphs. Plotkin et al. also applied this result to the partitioning of finite element method meshes in higher dimensions; for this generalization, shallow minors are necessary, as (with no depth restriction) the family of meshes in three or more dimensions has all graphs as minors. Teng (1998) extends these results to a broader class of high-dimensional graphs. More generally, a hereditary graph family has a separator theorem where the separator size is a sublinear power of n if and only if it has polynomial expansion.

Notes

References Dvořák, Zdeněk; Norin, Sergey (2015), Strongly sublinear separators and polynomial expansion, arXiv:1504.04821, Bibcode:2015arXiv150404821D. Plotkin, Serge; Rao, Satish; Smith, Warren D. (1994), "Shallow excluded minors and improved graph decompositions", Proc. 5th ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 462–470. Teng, Shang-Hua (1998), "Combinatorial aspects of geometric graphs", Comput. Geom., 9 (4): 277–287, doi:10.1016/S0925-7721(96)00008-9, MR 1609578. Wulff-Nilsen, Christian (2011), "Separator Theorems for Minor-Free and Shallow Minor-Free Graphs with Applications", Proc. 52nd IEEE Symp. Foundations of Computer Science (FOCS), pp. 37–46, arXiv:1107.1292, doi:10.1109/FOCS.2011.15, ISBN 978-0-7695-4571-4. Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28, Springer, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058.

Worked examples

Example 1 — a first encounter with Shallow minor

Start with the simplest possible case. Write down what Shallow minor 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 Shallow minor 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 Shallow minor 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 Shallow minor

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

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

Frequently asked questions

What is Shallow minor in simple terms?

Why does Shallow minor 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 Shallow minor?

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 Shallow minor.