Questions tagged [graph-reconstruction]

Questions related to graph reconstruction, the problem of reconstructing a graph from a set or deck (multiset) of subgraphs.

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Large family of subsets with small pairwise intersections

(Crossposting from StackExchange 4799692 after it has been there for a while.) Let $\alpha>0$ be a constant (can be sufficiently small if necessary) and $n$ be sufficiently large. What can we say ...
DesmondMiles's user avatar
4 votes
0 answers

Determined labels and reconstruction conjecture

In the reconstruction conjecture $(RC)$, if we know the $k$, $0\leq k \leq n$ labels of the vertices, then $RC$ is true. Do we know the minimum $k$ that the above statement is true? Do we have any ...
Shahrooz's user avatar
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4 votes
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From equivalent graphs to isomorphic one- Reconstruction Conjecture

Suppose we are working with connected simple graphs. We say the graph $G$ and $H$ are equivalent if for any spanning tree $T_G$ in $G$ there is an spanning tree $T_H$ in $H$ such that $T_G$ is ...
Shahrooz's user avatar
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3 votes
2 answers

Efficiently generating all regular/bidegreed graphs

There is a related question on how to generate all regular graphs; however, the procedure is random and repeats the generated graphs. Plus, there is no stop condition, unless recording the total ...
Sergey Ivanov's user avatar
4 votes
0 answers

Relation between two conjectures on reconstruction of graphs

In spectral graph theory, there is a conjecture that claims: Almost every graph is determined by its adjacency spectrum ($DS$). This conjecture belongs to professor Willem Haemers. Also, we have a ...
Shahrooz's user avatar
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Reconstruction conjecture:Complete graph invariants

Suppose we have a complete graph invariant $I$ i.e. for any two graphs $G$ and $H$ we have $ I(G)=I(H) \iff G \cong H $. Suppose now that the invariant $I$ is reconstructible from the desk of $G.$ ...
Leox's user avatar
  • 516
4 votes
1 answer

Edge Reconstruction Conjecture

I have seen this question asked at least once before, but not with any real answers. I was reading about the various reconstruction conjectures and equivalents, and I saw that the reconstruction ...
Brian Frazier's user avatar
13 votes
1 answer

Reconstruction Conjecture: are almost all digraphs reconstructible?

The Reconstruction Conjecture for simple graphs remains unresolved. Most attempts I've seen at resolving the conjecture aim at proving it to be true (or partially true). I don't believe there is a ...
Rebecca J. Stones's user avatar
1 vote
2 answers

edge graph reconstruction conjecture : set vs multi set

Why is the edge reconstruction conjecture stated with the deck defined as the multi set of graphs formed by deleting one edge? Can someone give an example of two graphs such that the edge deleted ...
Thinniyam Srinivasan Ramanatha's user avatar
3 votes
0 answers

Reconstructing a function from its variants that negate one argument

Call two functions $g(x_1,\ldots,x_n)$ and $h(x_1,\ldots,x_n)$ from complex numbers to complex numbers equivalent if they are the same up to the order of their arguments. Formally: there is a ...
Brendan McKay's user avatar
23 votes
0 answers

Is the Poset of Graphs Automorphism-free?

For $n\geq 5$, let $\mathcal {P}_n$ be the set of all isomorphism classes of graphs with n vertices. Give this set the poset structure given by $G \le H$ if and only if $G$ is a subgraph of $H$. Is ...
Wade Hann-Caruthers's user avatar
3 votes
2 answers

Less general edge reconstruction problem for simple graphs

Let $G$ be a simple graph. Let $E^-(G)$ denote the set of (isomorphism classes) of subgraphs of $G$ that can be obtained by deleting a single edge of $G$. Similarly, let $E^+(G)$ be the set of (...
anonon's user avatar
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5 votes
1 answer

Reconstructing the number of Hamiltonian cycles

As is common terminology in graph reconstruction, given a graph $G$, we call a vertex deleted subgraph of $G$, a card, and call the multiset of all cards, the deck of $G$. The graph reconstruction ...
Gjergji Zaimi's user avatar
4 votes
1 answer

If a graph invariant is NP-Hard, is its "deck ratio" NP-Hard as well?

This question is inspired by the Graph Reconstruction Conjecture. Suppose that $\psi$ is some graph invariant and that it is NP-Hard. There is a plethora of examples, of course. Now define $D_{\psi}(G)...
Felix Goldberg's user avatar
17 votes
1 answer

Reconstruction Conjecture holds for Directed Acyclic Graphs?

Wikipedia's article on the Reconstruction Conjecture mentions that the conjecture is false for digraphs, and refers to two papers by Stockmeyer. As far as I can see, none of the counter-examples in ...
Dag Oskar Madsen's user avatar
8 votes
5 answers

Reconstructing graphs with vertices of degree $k$ and $k-1$

The Graph Reconstruction Conjecture claims that any simple graph with 3 or more vertices is reconstructible from its "deck" of vertex-deleted subgraphs. (A nice introduction to this problem is at this ...
Ken W. Smith's user avatar
14 votes
0 answers

Reconstruction conjecture and partial 2-trees

Reconstruction conjecture says that graphs (with at least three vertices) are determined uniquely by their vertex deleted subgraphs. This conjecture is five decades old. Searching relevant literature,...
Shiva Kintali's user avatar
6 votes
1 answer

Reconstruction Conjecture: Group theoretic formulation

As we read from wiki, informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. Is there a group-theoretic formulation of this conjecture? ...
Unknown's user avatar
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10 votes
1 answer

Reconstruction conjecture: Can other decks do the job?

The standard reconstruction conjecture states that a graph is determined by its deck of vertex-deleted subgraphs. Question: Have other decks been investigated, finding out that only vertex-...
Hans-Peter Stricker's user avatar