Graph editdistance and hereditary properties

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Graph edit-distance and hereditary properties

• Noga Alon
• Uri Stav

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Graph edit-distance

• D the minimum number of edge deletions and/or
  additions needed in order to turn into a
  graph isomorphic to
• graph property
• the minimum distance from to
  a graph in

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Graph edit-distance

• D the minimum number of edge deletions and/or
  additions needed in order to turn into a
  graph isomorphic to
• graph property
• the minimum distance from to
  a graph in

Q Which graph on n vertices is the furthest
from satisfying ? Q What is the maximum
distance?
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Example triangle freeness

• triangle freeness
• is the furthest

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Monotone properties

• Closed under removal of vertices and edges
• Or closed under taking subgraphs
• Complete graphs
• By Erdos-Stone, Erdos-Simonovits

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Hereditary properties

• Closed under removal of vertices
• Or closed under taking induced subgraphs
• For example
• Monotone properties
• Induced H-freeness
• Perfect
• Chordal
• Cograph

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Related work

• Speed (size) of hereditary properties and the
  probability
• Prömel and Steger (induced H-freeness)
• Bollobás and Thomason (hereditary properties)
• Avoiding patterns in matrices
• Axenovich, Kézdy and Martin
• Testing hereditary graph properties
• Alon and Shapira
• Convergent graph sequences
• Lovász and Szegedy

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New result Hereditary properties
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Hereditary properties (cont.)
Theorem For any hereditary , there is such
that w.h.p. is the furthest from
up to

• E.g. - monotone properties ( )
• Possible proofs
• Szemerédi Regularity Lemma
• Convergent graph sequences LS
• Non-hereditary properties
• Example - being regular is close,
  is far

Proof sketch ?
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Colored regularity graphs

• A complete colored graph K
• Vertices black or white
• Edges black, white or grey
• V(K)v1,,vk
• G conforms to K if
• V1,,Vk a partition of V(G)
• Such that
• vi is black ? Vi is a clique
• vi is white ? Vi is an independent set
• (vi,vj) is black ? (Vi,Vj) is complete bipartite
• (vi,vj) is white ? (Vi,Vj) is edgeless
• (vi,vj) is grey ? no restriction on (Vi,Vj) !

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Cleaning a graph

• Every graph G is close to conforming to some
  (small) colored regularity graph
• Clean the graph
• Consider a (regular) partition
• Make every cluster homogeneous
• Make every pair either
• Complete bipartite
• Edgeless bipartite
• Very regular with fair edge density
• ? This clean graph defines a K to which it
  conforms

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Clean graphs key lemma

• Lemma (based on AS)
• is hereditary and ,
• Then there is a colored regularity graph K s.t.
  
• G is close to conforming to K
• any graph that conforms to K satisfies

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Proof sketch
is essentially the furthest graph
from P
conforming to K
The closest graph to in
the distance from conforming to K
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Well, then
is essentially the furthest graph from

• For any hereditary graph property,
• What is ?
• What is ?
• known for some families
• Sparse, Complement invariant,

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Forbidden induced subgraphs
is essentially the furthest graph
from P

• - induced H-freeness
• Given a graph H, what is
  ?
• AKM In some cases

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Example Induced C4-freeness

• being induced -free
• Split graphs
• Every split graph is induced -free

Observation
A
B
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Induced C4-freeness (cont.)
Lemma 1 Attained by

• ?

Lemma 2 Attained by
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Structural stability
What is the closest graph in to ?

• Critical forbidden induced subgraphs
• Non-stable forbidden H
• C5 , P4 (cographs), K1,2 ,

Theorem W.h.p. the closest induced C4 free
graph to G(n,½) is a split graph
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Colorings
K1,3
b
?
?

• (r,s) coloring
• A graph G is (r,s)-colorable if V(G)
• can be partitioned into rs sets such that
• r induce independent sets
• s induce cliques
• ? If G is (1,0)-col. or (0,2)-col. ? G is induced
  K1,3-free
• Provide upper bounds on for any
• The bounds are tight for

?
c
a
d
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Claw-free graphs
K1,3
?
?

• and
• Claim
• Proof For any graph G,
• If ? turn it into a (0,2)-colorable graph
• If ? turn it into a (1,0)-colorable graph

?
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Examples
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Future Directions

• Determine for every graph H
• Is there a nice function?
• Determine for other hereditary
  properties
• Finding the exact maximum edit-distance
• Hardness of edge-modification problems

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Thank You!