science

Meyniel graph

Meyniel 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 Meyniel graph rather than just read about it. In short: In graph theory, a Meyniel graph is a graph in which every odd cycle of length five or more has at least two chords (edges connecting non-consecutive vertices of the cycle). The chords may be uncrossed (as shown in the figure) or they may cross each other, as long as there are at least two of them.

Key takeaways

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

Reference excerpt

In graph theory, a Meyniel graph is a graph in which every odd cycle of length five or more has at least two chords (edges connecting non-consecutive vertices of the cycle). The chords may be uncrossed (as shown in the figure) or they may cross each other, as long as there are at least two of them. The Meyniel graphs are named after Henri Meyniel (also known for Meyniel's conjecture), who proved that they are perfect graphs in 1976, long before the proof of the strong perfect graph theorem completely characterized the perfect graphs. The same result was independently discovered by Markosjan & Karapetjan (1976).

Perfection The Meyniel graphs are a subclass of the perfect graphs. Every induced subgraph of a Meyniel graph is another Meyniel graph, and in every Meyniel graph the size of a maximum clique equals the minimum number of colors needed in a graph coloring. Thus, the Meyniel graphs meet the definition of being a perfect graph, that the clique number equals the chromatic number in every induced subgraph. Meyniel graphs are also called the very strongly perfect graphs, because (as Meyniel conjectured and Hoàng proved) they can be characterized by a property generalizing the defining property of the strongly perfect graphs: in every induced subgraph of a Meyniel graph, every vertex belongs to an independent set that intersects every maximal clique.

Related graph classes The Meyniel graphs contain the chordal graphs, the parity graphs, and their subclasses the interval graphs, distance-hereditary graphs, bipartite graphs, and line perfect graphs.

Although Meyniel graphs form a very general subclass of the perfect graphs, they do not include all perfect graphs. For instance the house graph (a pentagon with only one chord) is perfect but is not a Meyniel graph.

Algorithms and complexity Meyniel graphs can be recognized in polynomial time, and several graph optimization problems including graph coloring that are NP-hard for arbitrary graphs can be solved in polynomial time for Meyniel graphs.

References

Illustrations

Worked examples

Example 1 — a first encounter with Meyniel graph

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

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

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Meyniel graph in 20 minutes

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

Frequently asked questions

What is Meyniel graph in simple terms?

Why does Meyniel 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 Meyniel 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 Meyniel graph.