computer science

Graph sandwich problem

Graph sandwich problem is a computer 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 Graph sandwich problem rather than just read about it. In short: In graph theory and computer science, the graph sandwich problem is a problem of finding a graph that belongs to a particular family of graphs and is "sandwiched" between two other graphs, one of which must be a subgraph and the other of which must be a supergraph of the desired graph. Graph sandwich problems generalize the problem of testing whether a given graph belongs to a family of graphs, and have attracted at…

Key takeaways

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

Reference excerpt

In graph theory and computer science, the graph sandwich problem is a problem of finding a graph that belongs to a particular family of graphs and is "sandwiched" between two other graphs, one of which must be a subgraph and the other of which must be a supergraph of the desired graph. Graph sandwich problems generalize the problem of testing whether a given graph belongs to a family of graphs, and have attracted attention because of their applications and as a natural generalization of recognition problems.

Problem statement More precisely, given a vertex set V, a mandatory edge set E1, and a (potentially) larger edge set E2, a graph G = (V, E) is called a sandwich graph for the pair G1 = (V, E1), G2 = (V, E2) if E1 ⊆ E ⊆ E2. The graph sandwich problem for property Π is defined as follows:

Graph Sandwich Problem for Property Π: Instance: Vertex set V and edge sets E1 ⊆ E2 ⊆ V × V. Question: Is there a graph G = (V, E) such that E1 ⊆ E ⊆ E2 and G satisfies property Π ? The recognition problem for a class of graphs (those satisfying a property Π) is equivalent to the particular graph sandwich problem where E1 = E2, that is, the optional edge set is empty.

Computational complexity The graph sandwich problem is NP-complete when Π is the property of being a chordal graph, comparability graph, permutation graph, chordal bipartite graph, or chain graph. It can be solved in polynomial time for split graphs, threshold graphs, and graphs in which every five vertices contain at most one four-vertex induced path. The complexity status has also been settled for the H-free graph sandwich problems for each of the four-vertex graphs H.

References

Further reading Dantas, Simone; de Figueiredo, Celina M.H.; da Silva, Murilo V.G.; Teixeira, Rafael B. (2011), "On the forbidden induced subgraph sandwich problem", Discrete Applied Mathematics, 159 (16): 1717–1725, doi:10.1016/j.dam.2010.11.010.

Worked examples

Example 1 — a first encounter with Graph sandwich problem

Start with the simplest possible case. Write down what Graph sandwich problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer 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 Graph sandwich problem 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 Graph sandwich problem 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 Graph sandwich problem

In research: Graph sandwich problem appears in computer 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 Graph sandwich problem 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: Graph sandwich problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational problems in graph theory, NP-complete problems, so understanding it makes those chapters shorter.
In everyday life: Look for Graph sandwich problem 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 Graph sandwich problem in 20 minutes

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

Frequently asked questions

What is Graph sandwich problem in simple terms?

Why does Graph sandwich problem matter?

Because it connects several computer 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 Graph sandwich problem?

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 Graph sandwich problem.