science

Heawood family

Heawood family 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 Heawood family rather than just read about it. In short: In graph theory the term Heawood family refers to either one of the following two related graph families generated via ΔY- and YΔ-transformations: the family of 20 graphs generated from the complete graph K 7 {\displaystyle K_{7}} . the family of 78 graphs generated from K 7 {\displaystyle K_{7}} and K 3 , 3 , 1 , 1 {\displaystyle K_{3,3,1,1}} . In either setting the members of the graph family are collectively know…

Key takeaways

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

Reference excerpt

In graph theory the term Heawood family refers to either one of the following two related graph families generated via ΔY- and YΔ-transformations:

the family of 20 graphs generated from the complete graph K 7 {\displaystyle K_{7}} . the family of 78 graphs generated from K 7 {\displaystyle K_{7}} and K 3 , 3 , 1 , 1 {\displaystyle K_{3,3,1,1}} . In either setting the members of the graph family are collectively known as Heawood graphs, as the Heawood graph is a member. This is in analogy to the Petersen family, which too is named after its member the Petersen graph. The Heawood families are significant in topological graph theory. They contain the smallest known examples of intrinsically knotted graphs, of graphs that are not 4-flat, and of graphs with Colin de Verdière graph invariant μ = 6 {\displaystyle \mu =6} .

The K 7 {\displaystyle K_{7}} -family The K 7 {\displaystyle K_{7}} -family is generated from the complete graph K 7 {\displaystyle K_{7}} through repeated application of ΔY- and YΔ-transformations. The family consists of 20 graphs, all of which have 21 edges. The unique smallest member, K 7 {\displaystyle K_{7}} , has seven vertices. The unique largest member, the Heawood graph, has 14 vertices. Only 14 out of the 20 graphs are intrinsically knotted, all of which are minor minimal with this property. The other six graphs have knotless embeddings. This shows that knotless graphs are not closed under ΔY- and YΔ-transformations. All members of the K 7 {\displaystyle K_{7}} -family are intrinsically chiral.

The K 3 , 3 , 1 , 1 {\displaystyle K_{3,3,1,1}} -family The K 3 , 3 , 1 , 1 {\displaystyle K_{3,3,1,1}} -family is generated from the complete multipartite graph K 3 , 3 , 1 , 1 {\displaystyle K_{3,3,1,1}} through repeated application of ΔY- and YΔ-transformations. The family consists of 58 graphs, all of which have 22 edges. The unique smallest member, K 3 , 3 , 1 , 1 {\displaystyle K_{3,3,1,1}} , has eight vertices. The unique largest member has 14 vertices. All graphs in this family are intrinsically knotted and are minor minimal with this property.

The { K 7 , K 3 , 3 , 1 , 1 } {\displaystyle \{K_{7},K_{3,3,1,1}\}} -family

The Heawood family generated from both K 7 {\displaystyle K_{7}} and K 3 , 3 , 1 , 1 {\displaystyle K_{3,3,1,1}} through repeated application of ΔY- and YΔ-transformations is the disjoint union of the K 7 {\displaystyle K_{7}} -family and the K 3 , 3 , 1 , 1 {\displaystyle K_{3,3,1,1}} -family. It consists of 78 graphs. This graph family has significance in the study of 4-flat graphs, i.e., graphs with the property that every 2-dimensional CW complex built on them can be embedded into 4-space. Hein van der Holst (2006) showed that the graphs in the Heawood family are not 4-flat and have Colin de Verdière graph invariant μ = 6 {\displaystyle \mu =6} . In particular, they are neither planar nor linkless. Van der Holst suggested that they might form the complete list of excluded minors for both the 4-flat graphs and the graphs with μ ≤ 5 {\displaystyle \mu \leq 5} . This conjecture can be further motivated from structural similarities to other topologically defined graphs classes:

K 3 {\displaystyle K_{3}} and K 3 , 1 {\displaystyle K_{3,1}} (which generate each other, and no other simple graphs) are the excluded minors for linear forests and μ ≤ 1 {\displaystyle \mu \leq 1} .

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Heawood family

Start with the simplest possible case. Write down what Heawood family 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 Heawood family 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 Heawood family 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 Heawood family

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

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

Frequently asked questions

What is Heawood family in simple terms?

In graph theory the term Heawood family refers to either one of the following two related graph families generated via ΔY- and YΔ-transformations: the family of 20 graphs generated from the complete graph K 7 {\displaystyle K_{7}} . the family of 78 graphs generated from K 7 {\displaystyle K_{7}} a…

Why does Heawood family 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 Heawood family?

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 Heawood family.