Graph Reconstruction Numbers Brian McMullen

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Graph Reconstruction NumbersBrian McMullen
Stanislaw Radziszowski, Chair Christopher Homan,
Reader Darren Narayan, Observer
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Background

• Builds on project completed by Jennifer Baldwin
  during summer of 2004.
• Calculated reconstruction numbers for every
  finite, simple, undirected graph G where
  3V(G)8.
• Enhancements
• Extends calculations of reconstruction numbers to
  all graphs with nine or ten vertices.
• Examines causes of high existential
  reconstruction numbers.
• Tests 2-reconstructibility of graphs.

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Brief Review

• Complete graphs/cliques

• Graph complements

• Isomorphism

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Decks of Graphs

• Multiset of vertex deleted subgraphs of a graph.
• Example deck

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G-v0
G-v1
G-v2
G-v3
cards
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The Reconstruction Conjecture

• Proposed by P.J. Kelly and S.M. Ulam in 1942.
• States that every graph with three or more
  vertices is reconstructible up to isomorphism
  given its deck.
• In other words, D(G)D(H) if and only if G_at_ H.
• Graphs proven to be reconstructible.
• Regular graphs (Kelly, 1957)
• Disconnected graphs (Kelly, 1957)
• Trees (Kelly, 1957)

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k-Reconstructibility

• The deck of G where each card is created by
  deleting a unique set of k vertices is Dk(G).
• Some graph G is k-reconstructible if it is
  reconstructible up to isomorphism given Dk(G).
• The 2-reconstructibility for every graph G where
  4V(G)9 was tested for this project.

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Reconstruction Numbers

• In most cases, not all cards from a deck are
  required to reconstruct a graph.

Only G can be constructed from subdeck
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Types of Reconstruction Numbers

• Existential reconstruction number
• The existential reconstruction number of G,
  ?rn(G), is the minimum number of vertex-deleted
  subgraphs of G required to uniquely construct G
  up to isomorphism.
• Universal reconstruction number
• The universal reconstruction number of G, ?rn(G),
  is the minimum number n for which all possible
  subdecks of D(G) of size n uniquely construct G
  up to isomorphism.

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Knowns Regarding rn(G)

• For every graph G where V(G)³3, rn(G)2.

b
b
a
a

• In the probabilistic sense, almost all graphs can
  be reconstructed using three cards from their
  decks. (Bollobás, 1990)

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rn Values for Disconnected Graphs

• For every disconnected graph G where each
  component is not isomorphic, rn(G)3.(Myrvold -
  1989, Molina - 1995)

• For every disconnected graph G of the form pC,
  rn(G)V(C)2. (Myrvold - 1989)

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More rn Facts for Disconnected Graphs

• For every disconnected graph G of the form pKc,
  rn(G)c2. (Myrvold - 1989)

• For every disconnected graph G of the form pC
  where C is not a complete graph, rn(G)V(C).
  (Asciak/Lauri - 2002)

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Proof rn(pKc)c2

• D(pKc)(pc) Kc-1È (p-1)Kc

• Consider graph HKc1È Kc-1È (p-2)Kc

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Finding Reconstruction Numbers

• Given graph G, we find all graphs that share at
  least one card in their decks with D(G).
• Achieved by finding all one-vertex extension
  graphs for each card from D(G).
• Given graph F of order n, there are potentially
  2n one-vertex extensions of F. (Usually much less
  considering isomorphism).
• By comparing D(G) to cards of extension graphs,
  we can find reconstruction numbers for G.

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Software Tools

• Nauty package by Brendan McKay
• ASCII representation of graphs (graph6).
• Efficient tests for isomorphism between graphs.
• Efficient isomorph-free exhaustive generation of
  graphs.
• Condor package
• Installed on machines in RIT CS labs.
• Distributes computations among several machines.
• Used for parallel processing.

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Improvement to Old Algorithm

• Stop depending on pre-generated files for finding
  necessary decks and extension graphs.
• Use equivalence of reconstruction numbers between
  graph complements to save calculations.

• More efficient method for calculating universal
  reconstruction numbers.

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Graphs Analyzed

• Old Results
• Computed reconstruction numbers for every graph G
  where 3V(G)8.
• New Results
• Reconstruction number calculations for every
  graph G where 3V(G)10 and all regular graphs
  with eleven vertices.
• Tested 2-reconstructibility of every graph H
  where 4V(H)9.

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rn Counts
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Classes of Graphs with High rn Values

• Disconnected Graphs
• pKc (Myrvold)
• pC (Asciak/Lauri)
• Redundantly Connected Cycles
• Pairs of Complete Graphs Connected by 1-1 Edges

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Disconnected Graphs - pKc

• pKc (Myrvold)
• 2K2, 2K4, 4K2, 2K3, 3K2, 3K3, 2K5, 5K2 and their
  complements.
• All values matched c2.
• Does there exist a graph G where V(G) is prime
  such that rn(G)3?
• Since rn(pKc)c2, given any composite number n,
  there exists a graph G with n vertices such that
  rn(G)3.

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Disconnected Graphs - pC

• pC where C is not complete (Asciak/Lauri)

A
B
3A
2B,1A
2A,1B
3B
Above graphs prove rn(H)3.
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Side Note on pC Graphs

• Given connected graph F where all cards from D(F)
  are isomporphic, rn(pF)³rn(F).
• Proof
• All cards from D(pF) are isomorphic.
• Given graph H where D(H) shares rn(F)-1 cards
  with F, HÈ (p-1)F shares at least rn(F)-1 cards
  with pF.

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Graphs of Redundantly Connected Cycles
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Why?
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General RCC Graphs

• Graph RCC(n,l) is a graph with n cycles, each of
  length l that are redundantly connected.
• If graph GRCC(n,l), then rn(G)³ n2.
• Definition
• Take graph F with n cycles, each of length l.
• The cycles of F are labeled such that vc,i
  vc,i1 mod 1 where 0 i l-1.
• After making connections vc,i vd,i1 mod l where
  c¹ d, we get RCC(n,l).

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RCC Examples
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Overlapping GraphspKc Complements and RCCs

• Given graph G of the form Kc,c where 2c, G_at_
  RCC(c/2,4).
• Given graph G of the form Kc,c,c, G_at_ RCC(c,3).
• Explains why no RCC graphs were identified before
  RCC(2,5).

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Pairs of Complete Graphs with 1-1 Connections

• Examples

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Graphs in Class with High rn

• For every graph G of the form KcbKc where
  2b3.

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Proof

• Two different cards are in the decks of these
  graphs.
• Kcb-1Kc-1

• KcbKc-1

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Proof - Continued

• Given graph Kc1b-1 Kc-1
• Contains (c1)-(b-1) copies of A. (c-b2³3)
• Given graph Kc1bKc-1
• Contains b copies of A. (³2)
• Contains c-b1 copies of B. (³2)
• Given graph Kc1b1Kc-1
• Contains b1 copies of B. (³3)

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Lower Bound rn for Kcc-1Kc

• Given graph G of the form Kcc-1Kc where
  V(G)³6, rn(G)³ c.
• Proof
• D(G) contains 2(c-1) copies of Kcc-2Kc-1 (graph
  A) and 2 copies of Kcc-1Kc-1 (graph B).
• The deck of graph Kc1c-1Kc-1 contains c-1
  copies of A and 2 copies of B.

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Graph Exception
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"rn Counts
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2-Reconstructibility Results

• Tested for every graph G where 4V(G)9.
• 7 of 11 graphs with 4 vertices were
  2-nonreconstructible.
• 4 graphs with 5 vertices were 2-nonreconstructible
  .
• Every graph G where 6V(G)9 were found to be
  2-reconstructible.
• For every graph G where V(G)5, is G
  2-reconstructible?

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Four 2-nonreconstructible Graphs
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Further Considerations

• Does there exist a graph G with a prime number of
  vertices such that rn(G)3?
• Can we improve algorithm to calculate
  reconstruction nubers for all graphs with 11
  vertices?
• Are there more classes of graph with high rn?
• Does there exist a graph G where V(G)5 such
  that G is 2-nonreconstructible?
• How can we make better predictions for rn values?

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Questions?
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References

• K.J. Asciak and J. Lauri, On Disconnected Graph
  with Large Reconstruction Number. Ars
  Combinatoria, 62173-181, 2002.
• Baldwin, Jennifer. Graph Reconstruction Numbers.
  RIT Masters Project Report, 2004.
• B. Bollobás. Almost every graph has
  reconstruction number three. Jounal of Graph
  Theory, 14(1)1-4, 1990.
• Condor Team, University of Wisconsin-Madison.
  Condor Version 6.6.7 Manual, May 2004.
• P.J. Kelly, A congruence theorem for trees.
  Pacific Journal of Mathematics, 7961-968, 1957.
• B. McKay. Nauty Users Guide (Version 2.2).
  Computer Science Department, Australian National
  University, bdm_at_cs.anu.edu.au.
• R. Molina. Correction of a proof on the
  ally-reconstruction number of disconnected
  graphs, Ars Combinatoria, 2059-64, 1988.
• W. Myrvold. The ally-reconstruction number of a
  disconnected graph. Ars Combinatorics,
  28123-127, 1989.