Subgraph isomorphism problem

Key takeaways

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

Reference excerpt

In theoretical computer science, the subgraph isomorphism problem is a computational task in which two graphs G {\displaystyle G} and H {\displaystyle H} are given as input, and one must determine whether G {\displaystyle G} contains a subgraph that is isomorphic to H {\displaystyle H} . Subgraph isomorphism is a generalization of both the maximum clique problem and the problem of testing whether a graph contains a Hamiltonian cycle, and is therefore NP-complete. However certain other cases of subgraph isomorphism may be solved in polynomial time. Sometimes the name subgraph matching is also used for the same problem. This name puts emphasis on finding such a subgraph as opposed to the bare decision problem.

Decision problem and computational complexity

To prove subgraph isomorphism is NP-complete, it must be formulated as a decision problem. The input to the decision problem is a pair of graphs G {\displaystyle G} and H. The answer to the problem is positive if H is isomorphic to a subgraph of G, and negative otherwise. Formal question: Let G = ( V , E ) {\displaystyle G=(V,E)} , H = ( V ′ , E ′ ) {\displaystyle H=(V^{\prime },E^{\prime })} be graphs. Is there a subgraph G 0 = ( V 0 , E 0 ) ∣ V 0 ⊆ V , E 0 ⊆ E ∩ ( V 0 × V 0 ) {\displaystyle G_{0}=(V_{0},E_{0})\mid V_{0}\subseteq V,E_{0}\subseteq E\cap (V_{0}\times V_{0})} such that G 0 ≅ H {\displaystyle G_{0}\cong H} ? I. e., does there exist a bijection f : V 0 → V ′ {\displaystyle f\colon V_{0}\rightarrow V^{\prime }} such that { v 1 , v 2 } ∈ E 0 ⟺ { f ( v 1 ) , f ( v 2 ) } ∈ E ′ {\displaystyle \{\,v_{1},v_{2}\,\}\in E_{0}\iff \{\,f(v_{1}),f(v_{2})\,\}\in E^{\prime }} ? The proof of subgraph isomorphism being NP-complete is simple and based on reduction of the clique problem, an NP-complete decision problem in which the input is a single graph G and a number k, and the question is whether G contains a complete subgraph with k vertices. To translate this to a subgraph isomorphism problem, simply let H be the complete graph Kk; then the answer to the subgraph isomorphism problem for G and H is equal to the answer to the clique problem for G and k. Since the clique problem is NP-complete, this polynomial-time many-one reduction shows that subgraph isomorphism is also NP-complete. An alternative reduction from the Hamiltonian cycle problem translates a graph G which is to be tested for Hamiltonicity into the pair of graphs G and H, where H is a cycle having the same number of vertices as G. Because the Hamiltonian cycle problem is NP-complete even for planar graphs, this shows that subgraph isomorphism remains NP-complete even in the planar case. Subgraph isomorphism is a generalization of the graph isomorphism problem, which asks whether G is isomorphic to H: the answer to the graph isomorphism problem is true if and only if G and H both have the same numbers of vertices and edges and the subgraph isomorphism problem for G and H is true. However the complexity-theoretic status of graph isomorphism remains an open question. In the context of the Aanderaa–Karp–Rosenberg conjecture on the query complexity of monotone graph properties, Gröger (1992) showed that any subgraph isomorphism problem has query complexity Ω(n3/2); that is, solving the subgraph isomorphism requires an algorithm to check the presence or absence in the input of Ω(n3/2) different edges in the graph.

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Illustrations

Worked examples

Example 1 — a first encounter with Subgraph isomorphism problem

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

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

How to study Subgraph isomorphism problem in 20 minutes

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

Frequently asked questions

What is Subgraph isomorphism problem in simple terms?

In theoretical computer science, the subgraph isomorphism problem is a computational task in which two graphs G {\displaystyle G} and H {\displaystyle H} are given as input, and one must determine whether G {\displaystyle G} contains a subgraph that is isomorphic to H {\displaystyle H} . Subgraph i…

Why does 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 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 Subgraph isomorphism problem.