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New digraph reconstruction conjecture

New digraph reconstruction conjecture 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 New digraph reconstruction conjecture rather than just read about it. In short: The reconstruction conjecture of Stanisław Ulam is one of the best-known open problems in graph theory. Using the terminology of Frank Harary it can be stated as follows: If G and H are two graphs on at least three vertices and ƒ is a bijection from V(G) to V(H) such that G\{v} and H\{ƒ(v)} are isomorphic for all vertices v in V(G), then G and H are isomorphic.

New digraph reconstruction conjecture — main illustration
New digraph reconstruction conjecture — illustration

Key takeaways

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

Reference excerpt

The reconstruction conjecture of Stanisław Ulam is one of the best-known open problems in graph theory. Using the terminology of Frank Harary it can be stated as follows: If G and H are two graphs on at least three vertices and ƒ is a bijection from V(G) to V(H) such that G\{v} and H\{ƒ(v)} are isomorphic for all vertices v in V(G), then G and H are isomorphic. In 1964 Harary extended the reconstruction conjecture to directed graphs on at least five vertices as the so-called digraph reconstruction conjecture. Many results supporting the digraph reconstruction conjecture appeared between 1964 and 1976. However, this conjecture was proved to be false when P. K. Stockmeyer discovered several infinite families of counterexample pairs of digraphs (including tournaments) of arbitrarily large order. The falsity of the digraph reconstruction conjecture caused doubt about the reconstruction conjecture itself. Stockmeyer even observed that “perhaps the considerable effort being spent in attempts to prove the (reconstruction) conjecture should be balanced by more serious attempts to construct counterexamples.” In 1979, Ramachandran revived the digraph reconstruction conjecture in a slightly weaker form called the new digraph reconstruction conjecture. In a digraph, the number of arcs incident from (respectively, to) a vertex v is called the outdegree (respectively, indegree) of v and is denoted by od(v) (respectively, id(v)). The new digraph conjecture may be stated as follows:

If D and E are any two digraphs and ƒ is a bijection from V(D) to V(E) such that D\{v} and E\{ƒ(v)} are isomorphic and (od(v),id(v)) = (od(ƒ(v)),id(ƒ(v))) for all v in V(D), then D and E are isomorphic.

The new digraph reconstruction conjecture reduces to the reconstruction conjecture in the undirected case, because if all the vertex-deleted subgraphs of two graphs are isomorphic, then the corresponding vertices must have the same degree. Thus, the new digraph reconstruction conjecture is stronger than the reconstruction conjecture, but weaker than the disproved digraph reconstruction conjecture. Several families of digraphs have been shown to satisfy the new digraph reconstruction conjecture and these include all the digraphs in the known counterexample pairs to the digraph reconstruction conjecture.

Reductions All digraphs are N-reconstructible if all digraphs with 2-connected underlying graphs are N-reconstructible. All digraphs are N-reconstructible if and only if either of the following two classes of digraphs are N-reconstructible, where diam(D) and radius(D) are defined to be the diameter and radius of the underlying graph of D. Digraphs with diam(D) ≤ 2 or diam(D) = diam(Dc) = 3 Digraphs D with 2-connected underlying graphs and radius(D) ≤ 2

Present status As of 2026, no counterexample to the new digraph reconstruction conjecture is known. This conjecture is now also known as the degree associated reconstruction conjecture.

References

Illustrations

New digraph reconstruction conjecture: Each vertex in graph 1 matches one from graph 2. In each subgraph made by removing one of the vertices from graph 1, and the matching vertex from graph 2, the outdegree of each remaining vertex in graph 1's subgraph is the same as its matching counterpart in graph 2's subgraph. This will always be true given graphs 1 and 2 are isomorphic, but the conjecture states that this works in reverse, where knowing only that vertices between two graphs can be paired such that the outdegrees in each subgraph constructed as described match for every vertex is enough information to determine the two graphs are isomorphic.
Each vertex in graph 1 matches one from graph 2. In each subgraph made by removing one of the vertices from graph 1, and the matching vertex from graph 2, the outdegree of each remaining vertex in graph 1's subgraph is the same as its matching counterpart in graph 2's subgraph. This will always be true given graphs 1 and 2 are isomorphic, but the conjecture states that this works in reverse, where knowing only that vertices between two graphs can be paired such that the outdegrees in each subgraph constructed as described match for every vertex is enough information to determine the two graphs are isomorphic.

Worked examples

Example 1 — a first encounter with New digraph reconstruction conjecture

Start with the simplest possible case. Write down what New digraph reconstruction conjecture 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 New digraph reconstruction conjecture 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 New digraph reconstruction conjecture 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 New digraph reconstruction conjecture

In research
New digraph reconstruction conjecture 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 New digraph reconstruction conjecture 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
New digraph reconstruction conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Conjectures, Directed graphs, Unsolved problems in graph theory, so understanding it makes those chapters shorter.
In everyday life
Look for New digraph reconstruction conjecture 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 New digraph reconstruction conjecture in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what New digraph reconstruction conjecture means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain New digraph reconstruction conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is New digraph reconstruction conjecture in simple terms?

The reconstruction conjecture of Stanisław Ulam is one of the best-known open problems in graph theory. Using the terminology of Frank Harary it can be stated as follows: If G and H are two graphs on at least three vertices and ƒ is a bijection from V(G) to V(H) such that G\{v} and H\{ƒ(v)} are iso…

Why does New digraph reconstruction conjecture 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 New digraph reconstruction conjecture?

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 New digraph reconstruction conjecture.

Tags

  • Conjectures
  • Directed graphs
  • Unsolved problems in graph theory

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