Graph reconstruction from
multi-vertex-deleted subgraphs
Joint work with James Tuite
Grahame Erskine
The Open University, UK
Scottish Combinatorics Meeting , Glasgow, 27 April 2016
The main problem
Given a graph of order n, form the n induced subgraphs of order n − 1.
The main problem
Given a graph of order n, form the n induced subgraphs of order n − 1.
The main problem
Given a graph of order n, form the n induced subgraphs of order n − 1.
Q: Is this process reversible?
The main problem
Given a graph of order n, form the n induced subgraphs of order n − 1.
Q: Is this process reversible?
Conjecture (Kelly/Ulam, c. 1941):
All graphs of order at least 3 are reconstructible.
Example
These are the single vertex deleted subgraphs of a graph of order six.
Example
These are the single vertex deleted subgraphs of a graph of order six.
What was the original graph?
What is known?
What is known?
• If |V(H)| < |V(G)| then the number of subgraphs isomorphic to H
contained in G is deducible from the deck of G. (Kelly, 1957)
What is known?
• If |V(H)| < |V(G)| then the number of subgraphs isomorphic to H
contained in G is deducible from the deck of G. (Kelly, 1957)
• Hence the number of edges and therefore the degree sequence of
a graph is deducible from its deck.
What is known?
• If |V(H)| < |V(G)| then the number of subgraphs isomorphic to H
contained in G is deducible from the deck of G. (Kelly, 1957)
• Hence the number of edges and therefore the degree sequence of
a graph is deducible from its deck.
• Hence all regular graphs are reconstructible (Kelly, 1957)
What is known?
• If |V(H)| < |V(G)| then the number of subgraphs isomorphic to H
contained in G is deducible from the deck of G. (Kelly, 1957)
• Hence the number of edges and therefore the degree sequence of
a graph is deducible from its deck.
• Hence all regular graphs are reconstructible (Kelly, 1957)
• G is reconstructible if and only if G is reconstructible.
What is known?
• If |V(H)| < |V(G)| then the number of subgraphs isomorphic to H
contained in G is deducible from the deck of G. (Kelly, 1957)
• Hence the number of edges and therefore the degree sequence of
a graph is deducible from its deck.
• Hence all regular graphs are reconstructible (Kelly, 1957)
• G is reconstructible if and only if G is reconstructible.
• All disconnected graphs are reconstructible (Kelly, 1957)
What is known?
• If |V(H)| < |V(G)| then the number of subgraphs isomorphic to H
contained in G is deducible from the deck of G. (Kelly, 1957)
• Hence the number of edges and therefore the degree sequence of
a graph is deducible from its deck.
• Hence all regular graphs are reconstructible (Kelly, 1957)
• G is reconstructible if and only if G is reconstructible.
• All disconnected graphs are reconstructible (Kelly, 1957)
• All trees are reconstructible (Bondy, Hemminger, 1977)
What is known?
• If |V(H)| < |V(G)| then the number of subgraphs isomorphic to H
contained in G is deducible from the deck of G. (Kelly, 1957)
• Hence the number of edges and therefore the degree sequence of
a graph is deducible from its deck.
• Hence all regular graphs are reconstructible (Kelly, 1957)
• G is reconstructible if and only if G is reconstructible.
• All disconnected graphs are reconstructible (Kelly, 1957)
• All trees are reconstructible (Bondy, Hemminger, 1977)
• Almost all graphs are reconstructible (Bollobás, 1990)
What is known?
• If |V(H)| < |V(G)| then the number of subgraphs isomorphic to H
contained in G is deducible from the deck of G. (Kelly, 1957)
• Hence the number of edges and therefore the degree sequence of
a graph is deducible from its deck.
• Hence all regular graphs are reconstructible (Kelly, 1957)
• G is reconstructible if and only if G is reconstructible.
• All disconnected graphs are reconstructible (Kelly, 1957)
• All trees are reconstructible (Bondy, Hemminger, 1977)
• Almost all graphs are reconstructible (Bollobás, 1990)
• All graphs of order at most 11 are reconstructible (McKay, 1997)
Generalisations
We can generalise this question in many ways.
Generalisations
We can generalise this question in many ways.
1. What happens if we delete k vertices
at a time instead of just one,
i.e. the deck consists of the nk induced subgraphs of order n − k?
Generalisations
We can generalise this question in many ways.
1. What happens if we delete k vertices
at a time instead of just one,
i.e. the deck consists of the nk induced subgraphs of order n − k?
2. What happens if we ignore multiplicities of isomorphs in the deck
(the set reconstruction problem)?
Generalisations
We can generalise this question in many ways.
1. What happens if we delete k vertices
at a time instead of just one,
i.e. the deck consists of the nk induced subgraphs of order n − k?
2. What happens if we ignore multiplicities of isomorphs in the deck
(the set reconstruction problem)? Conjecture: all graphs of order
at least 4 are set-reconstructible.
Generalisations
We can generalise this question in many ways.
1. What happens if we delete k vertices
at a time instead of just one,
i.e. the deck consists of the nk induced subgraphs of order n − k?
2. What happens if we ignore multiplicities of isomorphs in the deck
(the set reconstruction problem)? Conjecture: all graphs of order
at least 4 are set-reconstructible.
3. When are other graph invariants such as adjacency or Laplacian
spectrum reconstructible from the deck?
Generalisations
We can generalise this question in many ways.
1. What happens if we delete k vertices
at a time instead of just one,
i.e. the deck consists of the nk induced subgraphs of order n − k?
2. What happens if we ignore multiplicities of isomorphs in the deck
(the set reconstruction problem)? Conjecture: all graphs of order
at least 4 are set-reconstructible.
3. When are other graph invariants such as adjacency or Laplacian
spectrum reconstructible from the deck?
4. If a graph is reconstructible, can we find a small subset of its deck
which still uniquely determines it?
Generalisations
We can generalise this question in many ways.
1. What happens if we delete k vertices
at a time instead of just one,
i.e. the deck consists of the nk induced subgraphs of order n − k?
2. What happens if we ignore multiplicities of isomorphs in the deck
(the set reconstruction problem)? Conjecture: all graphs of order
at least 4 are set-reconstructible.
3. When are other graph invariants such as adjacency or Laplacian
spectrum reconstructible from the deck?
4. If a graph is reconstructible, can we find a small subset of its deck
which still uniquely determines it?
5. What about reconstruction from edge-deleted subgraphs?
Non-reconstructible graphs
Reconstructibility under k-vertex deletion:
Graph
order
3
4
5
6
7
8
9
10
11
Isomorphism
classes
4
11
34
156
1044
12346
274668
12005168
1018997864
k=1
0
0
0
0
0
0
0
0
0
k=2
4
7
4
0
0
0
0
Non-reconstructible graphs
k=3 k=4 k=5
k=6
–
–
–
–
11
–
–
–
30
34
–
–
78
152
156
–
20
854
1040
1044
8
1937 11935 12342
0
273846
k=7
–
–
–
–
–
12346
274664
Source: Rivshin & Radziszowski, Multi-vertex deletion graph reconstruction
numbers, Combin. Math. Combin. Comput. 78 (2011), 303-321.
Algorithms
Rivshin & Radziszowski (“one at a time”):
Algorithms
Rivshin & Radziszowski (“one at a time”):
• Start with a given graph G and compute its k-vertex-deleted deck.
Algorithms
Rivshin & Radziszowski (“one at a time”):
• Start with a given graph G and compute its k-vertex-deleted deck.
• For the first card in the deck, what possible graphs could it have
come from? (A small “universe of possible non-reconstructible
pairs”.)
Algorithms
Rivshin & Radziszowski (“one at a time”):
• Start with a given graph G and compute its k-vertex-deleted deck.
• For the first card in the deck, what possible graphs could it have
come from? (A small “universe of possible non-reconstructible
pairs”.)
• Do any of these have the same deck as G?
Algorithms
Rivshin & Radziszowski (“one at a time”):
• Start with a given graph G and compute its k-vertex-deleted deck.
• For the first card in the deck, what possible graphs could it have
come from? (A small “universe of possible non-reconstructible
pairs”.)
• Do any of these have the same deck as G?
Erskine & Tuite (“all at once”):
Algorithms
Rivshin & Radziszowski (“one at a time”):
• Start with a given graph G and compute its k-vertex-deleted deck.
• For the first card in the deck, what possible graphs could it have
come from? (A small “universe of possible non-reconstructible
pairs”.)
• Do any of these have the same deck as G?
Erskine & Tuite (“all at once”):
• Start with all graphs of given degree sequence. (A large “universe
of possible non-reconstructible pairs”.)
Algorithms
Rivshin & Radziszowski (“one at a time”):
• Start with a given graph G and compute its k-vertex-deleted deck.
• For the first card in the deck, what possible graphs could it have
come from? (A small “universe of possible non-reconstructible
pairs”.)
• Do any of these have the same deck as G?
Erskine & Tuite (“all at once”):
• Start with all graphs of given degree sequence. (A large “universe
of possible non-reconstructible pairs”.)
• Compute the k-deck for each using nauty to get isomorphism
class for each card.
Algorithms
Rivshin & Radziszowski (“one at a time”):
• Start with a given graph G and compute its k-vertex-deleted deck.
• For the first card in the deck, what possible graphs could it have
come from? (A small “universe of possible non-reconstructible
pairs”.)
• Do any of these have the same deck as G?
Erskine & Tuite (“all at once”):
• Start with all graphs of given degree sequence. (A large “universe
of possible non-reconstructible pairs”.)
• Compute the k-deck for each using nauty to get isomorphism
class for each card.
• Compute an 8 byte hash of each deck.
Algorithms
Rivshin & Radziszowski (“one at a time”):
• Start with a given graph G and compute its k-vertex-deleted deck.
• For the first card in the deck, what possible graphs could it have
come from? (A small “universe of possible non-reconstructible
pairs”.)
• Do any of these have the same deck as G?
Erskine & Tuite (“all at once”):
• Start with all graphs of given degree sequence. (A large “universe
of possible non-reconstructible pairs”.)
• Compute the k-deck for each using nauty to get isomorphism
class for each card.
• Compute an 8 byte hash of each deck.
• Once done, examine each hash collision to see if it was genuinely
a duplicate.
Non-reconstructible graphs
Our algorithm for finding non-reconstructible pairs works by computing
the deck for all graphs of a given order, then finding duplicates. This
uses a large amount of memory, but is quicker than previous methods.
Non-reconstructible graphs
Our algorithm for finding non-reconstructible pairs works by computing
the deck for all graphs of a given order, then finding duplicates. This
uses a large amount of memory, but is quicker than previous methods.
Graph
order
3
4
5
6
7
8
9
10
11
12
Isomorphism
classes
4
11
34
156
1044
12346
274668
12005168
1018997864
165091172592
k=1
0
0
0
0
0
0
0
0
0
0
k=2
4
7
4
0
0
0
0
0
0
Non-reconstructible graphs
k=3 k=4
k=5
k=6
–
–
–
–
11
–
–
–
30
34
–
–
78
152
156
–
20
854
1040
1044
8
1937 11935
12342
0
502 111338 273846
0
44
48674 8386570
0
0
5640
Source: E. & Tuite, 2016, unpublished.
k=7
–
–
–
–
–
12346
274664
12003580
How does k-vertex reconstructibility behave?
It is known (Nýdl, 1981) that for any k there is a non-k-reconstructible
graph of order 2k. Rivshin & Radziszowski (2011) make a number of
further conjectures.
How does k-vertex reconstructibility behave?
It is known (Nýdl, 1981) that for any k there is a non-k-reconstructible
graph of order 2k. Rivshin & Radziszowski (2011) make a number of
further conjectures.
1. For any k, there is some N(k) for which all graphs of order N(k) are
k-reconstructible.
How does k-vertex reconstructibility behave?
It is known (Nýdl, 1981) that for any k there is a non-k-reconstructible
graph of order 2k. Rivshin & Radziszowski (2011) make a number of
further conjectures.
1. For any k, there is some N(k) for which all graphs of order N(k) are
k-reconstructible.
2. For any k, if all graphs of order N are k-reconstructible then so are
all graphs of order n > N.
How does k-vertex reconstructibility behave?
It is known (Nýdl, 1981) that for any k there is a non-k-reconstructible
graph of order 2k. Rivshin & Radziszowski (2011) make a number of
further conjectures.
1. For any k, there is some N(k) for which all graphs of order N(k) are
k-reconstructible.
2. For any k, if all graphs of order N are k-reconstructible then so are
all graphs of order n > N.
3. For any k, the minimum value of N(k) is 3k.
How does k-vertex reconstructibility behave?
It is known (Nýdl, 1981) that for any k there is a non-k-reconstructible
graph of order 2k. Rivshin & Radziszowski (2011) make a number of
further conjectures.
1. For any k, there is some N(k) for which all graphs of order N(k) are
k-reconstructible.
2. For any k, if all graphs of order N are k-reconstructible then so are
all graphs of order n > N.
3. For any k, the minimum value of N(k) is 3k.
Our new data do not appear to suggest conjectures 1 and 2 are in
doubt.
How does k-vertex reconstructibility behave?
It is known (Nýdl, 1981) that for any k there is a non-k-reconstructible
graph of order 2k. Rivshin & Radziszowski (2011) make a number of
further conjectures.
1. For any k, there is some N(k) for which all graphs of order N(k) are
k-reconstructible.
2. For any k, if all graphs of order N are k-reconstructible then so are
all graphs of order n > N.
3. For any k, the minimum value of N(k) is 3k.
Our new data do not appear to suggest conjectures 1 and 2 are in
doubt.
Unfortunately, all graphs of order 11 are 4-reconstructible.
How does k-vertex reconstructibility behave?
It is known (Nýdl, 1981) that for any k there is a non-k-reconstructible
graph of order 2k. Rivshin & Radziszowski (2011) make a number of
further conjectures.
1. For any k, there is some N(k) for which all graphs of order N(k) are
k-reconstructible.
2. For any k, if all graphs of order N are k-reconstructible then so are
all graphs of order n > N.
3. For any k, the minimum value of N(k) is 3k.
Our new data do not appear to suggest conjectures 1 and 2 are in
doubt.
Unfortunately, all graphs of order 11 are 4-reconstructible.
Even more unfortunately:
Theorem (Nýdl, 1991): for any ε > 0 there exists a graph G which is
not ε|V(G)|-reconstructible.
Set non-reconstructible graphs
The same algorithm can be used for the set version of the problem:
Set non-reconstructible graphs
The same algorithm can be used for the set version of the problem:
Graph
order
3
4
5
6
7
8
9
10
11
Isomorphism
classes
4
11
34
156
1044
12346
274668
12005168
1018997864
k=1
2
0
0
0
0
0
0
0
0
k=2
4
9
29
62
64
0
0
4
2
Non-reconstructible graphs
k=3
k=4
k=5
k=6
–
–
–
–
11
–
–
–
32
34
–
–
154
154
156
–
948
1042
1042
1044
3586 12267
12344
12344
348 232977
274605
274666
160
48916 11940308 12005126
60
Source: E. & Tuite, 2016, unpublished.
k=7
–
–
–
–
–
12346
274666
12005166
Further questions
Further questions
1. The non-reconstructible pairs in the set problem at
k = 2, n = 10, 11 are intriguing. The graphs have different numbers
of edges. Can we understand this behaviour?
Further questions
1. The non-reconstructible pairs in the set problem at
k = 2, n = 10, 11 are intriguing. The graphs have different numbers
of edges. Can we understand this behaviour?
2. Is it possible that the set conjecture might be in trouble for k = 1
and large n?
Further questions
1. The non-reconstructible pairs in the set problem at
k = 2, n = 10, 11 are intriguing. The graphs have different numbers
of edges. Can we understand this behaviour?
2. Is it possible that the set conjecture might be in trouble for k = 1
and large n?
3. Can we combine this algorithm with known theory on
reconstructible graphs to push beyond n = 12 for some classes?
Further questions
1. The non-reconstructible pairs in the set problem at
k = 2, n = 10, 11 are intriguing. The graphs have different numbers
of edges. Can we understand this behaviour?
2. Is it possible that the set conjecture might be in trouble for k = 1
and large n?
3. Can we combine this algorithm with known theory on
reconstructible graphs to push beyond n = 12 for some classes?
Thanks for listening!