computer science

Induced subgraph isomorphism problem

Induced subgraph isomorphism 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 Induced subgraph isomorphism problem rather than just read about it. In short: In complexity theory and graph theory, induced subgraph isomorphism is an NP-complete decision problem that involves finding a given graph as an induced subgraph of a larger graph. Problem statement Formally, the problem takes as input two graphs G1=(V1, E1) and G2=(V2, E2), where the number of vertices in V1 can be assumed to be less than or equal to the number of vertices in V2.

Key takeaways

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

Reference excerpt

In complexity theory and graph theory, induced subgraph isomorphism is an NP-complete decision problem that involves finding a given graph as an induced subgraph of a larger graph.

Problem statement Formally, the problem takes as input two graphs G1=(V1, E1) and G2=(V2, E2), where the number of vertices in V1 can be assumed to be less than or equal to the number of vertices in V2. G1 is isomorphic to an induced subgraph of G2 if there is an injective function f which maps the vertices of G1 to vertices of G2 such that for all pairs of vertices x, y in V1, edge (x, y) is in E1 if and only if the edge (f(x), f(y)) is in E2. The answer to the decision problem is yes if this function f exists, and no otherwise. This is different from the subgraph isomorphism problem in that the absence of an edge in G1 implies that the corresponding edge in G2 must also be absent. In subgraph isomorphism, these "extra" edges in G2 may be present.

Computational complexity The complexity of induced subgraph isomorphism separates outerplanar graphs from their generalization series–parallel graphs: it may be solved in polynomial time for 2-connected outerplanar graphs, but is NP-complete for 2-connected series–parallel graphs.

Special cases The special case of finding a long path as an induced subgraph of a hypercube has been particularly well-studied, and is called the snake-in-the-box problem. The maximum independent set problem is also an induced subgraph isomorphism problem in which one seeks to find a large independent set as an induced subgraph of a larger graph, and the maximum clique problem is an induced subgraph isomorphism problem in which one seeks to find a large clique graph as an induced subgraph of a larger graph.

Differences with the subgraph isomorphism problem Although the induced subgraph isomorphism problem seems only slightly different from the subgraph isomorphism problem, the "induced" restriction introduces changes large enough that we can witness differences from a computational complexity point of view. For example, the subgraph isomorphism problem is NP-complete on connected proper interval graphs and on connected bipartite permutation graphs, but the induced subgraph isomorphism problem can be solved in polynomial time on these two classes. Moreover, the induced subtree isomorphism problem (i.e. the induced subgraph isomorphism problem where G1 is restricted to be a tree) can be solved in polynomial time on interval graphs, while the subtree isomorphism problem is NP-complete on proper interval graphs.

References

Illustrations

Worked examples

Example 1 — a first encounter with Induced subgraph isomorphism problem

Start with the simplest possible case. Write down what Induced subgraph isomorphism 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 Induced subgraph isomorphism 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 Induced subgraph isomorphism 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 Induced subgraph isomorphism problem

In research: Induced subgraph isomorphism 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 Induced subgraph isomorphism 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: Induced subgraph isomorphism 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 Induced subgraph isomorphism 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.

Affiliate

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

How to study Induced subgraph isomorphism problem in 20 minutes

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

Frequently asked questions

What is Induced subgraph isomorphism problem in simple terms?

In complexity theory and graph theory, induced subgraph isomorphism is an NP-complete decision problem that involves finding a given graph as an induced subgraph of a larger graph. Problem statement Formally, the problem takes as input two graphs G1=(V1, E1) and G2=(V2, E2), where the number of ver…

Why does Induced subgraph isomorphism 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 Induced subgraph isomorphism 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 Induced subgraph isomorphism problem.