1 extremal graph theory - iowa state university · 2008-10-27 · 1 extremal graph theory 2 edit...
TRANSCRIPT
![Page 1: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/1.jpg)
1 Extremal graph theory
2 Edit Distance Origins
3 Definitions related to Edit Distance
4 Hereditary Properties
5 Binary Chromatic Number
6 The Edit Distance Function
7 Weighted Turan lemma
8 Future Work
9 Thanks!
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 2 / 19
![Page 2: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/2.jpg)
A classical extremal problem
Suppose we have an n-vertex graph G .
We want to compute the fewest number of edge-additions plusedge-deletions to apply to G so that the resulting graph H has no triangles.
Theorem (Mantel, 1907)
If an n-vertex graph H has no triangles, then the number of edges H has isat most
⌊n2/4
⌋.
This bound is only achieved if H is complete bipartite.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 3 / 19
![Page 3: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/3.jpg)
A classical extremal problem
Suppose we have an n-vertex graph G .
We want to compute the fewest number of edge-additions plusedge-deletions to apply to G so that the resulting graph H has no triangles.
Theorem (Mantel, 1907)
If an n-vertex graph H has no triangles, then the number of edges H has isat most
⌊n2/4
⌋.
This bound is only achieved if H is complete bipartite.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 3 / 19
![Page 4: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/4.jpg)
A classical extremal problem
Suppose we have an n-vertex graph G .
We want to compute the fewest number of edge-additions plusedge-deletions to apply to G so that the resulting graph H has no triangles.
Theorem (Mantel, 1907)
If an n-vertex graph H has no triangles, then the number of edges H has isat most
⌊n2/4
⌋.
This bound is only achieved if H is complete bipartite.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 3 / 19
![Page 5: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/5.jpg)
A classical extremal problem
Suppose we have an n-vertex graph G .
We want to compute the fewest number of edge-additions plusedge-deletions to apply to G so that the resulting graph H has no triangles.
Theorem (Mantel, 1907)
If an n-vertex graph H has no triangles, then the number of edges H has isat most
⌊n2/4
⌋.
This bound is only achieved if H is complete bipartite.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 3 / 19
![Page 6: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/6.jpg)
Editing away triangles
So, if G were the complete graph, it would require at least(n
2
)−⌊n2/4
⌋
=
(bn/2c
2
)+
(dn/2e
2
)
edge-deletions.
But for any graph G , we can delete at most this many edges and remaintriangle-free:
Partition the vertices in half and delete edges inside each part.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 4 / 19
![Page 7: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/7.jpg)
Editing away triangles
So, if G were the complete graph, it would require at least(n
2
)−⌊n2/4
⌋=
(bn/2c
2
)+
(dn/2e
2
)edge-deletions.
But for any graph G , we can delete at most this many edges and remaintriangle-free:
Partition the vertices in half and delete edges inside each part.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 4 / 19
![Page 8: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/8.jpg)
Editing away triangles
So, if G were the complete graph, it would require at least(n
2
)−⌊n2/4
⌋=
(bn/2c
2
)+
(dn/2e
2
)edge-deletions.
But for any graph G , we can delete at most this many edges and remaintriangle-free:
Partition the vertices in half and delete edges inside each part.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 4 / 19
![Page 9: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/9.jpg)
Editing away triangles
So, if G were the complete graph, it would require at least(n
2
)−⌊n2/4
⌋=
(bn/2c
2
)+
(dn/2e
2
)edge-deletions.
But for any graph G , we can delete at most this many edges and remaintriangle-free:
Partition the vertices in half and delete edges inside each part.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 4 / 19
![Page 10: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/10.jpg)
Results on triangles
So, the maximum number of changes required to remove triangles fromn-vertex graph G is (
bn/2c2
)+
(dn/2e
2
)
∼ n2
4
.
This is achieved by G = Kn.
Instead of triangles, we can generalize the previous analysis to forbidcopies of K`+1, ` ≥ 2.
Theorem (Turan, 1941)
If an n-vertex graph H has no copy of K`, then the number of edges H hasis at most
n2
2`.
This bound is only achieved if H is complete `-partite and ` | n.
So, the maximum number of changes required to remove triangles fromn-vertex graph G is
∼ n2
2`
∼ 1
`
(n
2
).
This is achieved by G = Kn.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 5 / 19
![Page 11: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/11.jpg)
Results on triangles
So, the maximum number of changes required to remove triangles fromn-vertex graph G is (
bn/2c2
)+
(dn/2e
2
)∼ n2
4.
This is achieved by G = Kn.
Instead of triangles, we can generalize the previous analysis to forbidcopies of K`+1, ` ≥ 2.
Theorem (Turan, 1941)
If an n-vertex graph H has no copy of K`, then the number of edges H hasis at most
n2
2`.
This bound is only achieved if H is complete `-partite and ` | n.
So, the maximum number of changes required to remove triangles fromn-vertex graph G is
∼ n2
2`
∼ 1
`
(n
2
).
This is achieved by G = Kn.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 5 / 19
![Page 12: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/12.jpg)
Results on triangles
So, the maximum number of changes required to remove triangles fromn-vertex graph G is (
bn/2c2
)+
(dn/2e
2
)∼ n2
4.
This is achieved by G = Kn.
Instead of triangles, we can generalize the previous analysis to forbidcopies of K`+1, ` ≥ 2.
Theorem (Turan, 1941)
If an n-vertex graph H has no copy of K`, then the number of edges H hasis at most
n2
2`.
This bound is only achieved if H is complete `-partite and ` | n.
So, the maximum number of changes required to remove triangles fromn-vertex graph G is
∼ n2
2`
∼ 1
`
(n
2
).
This is achieved by G = Kn.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 5 / 19
![Page 13: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/13.jpg)
Results on triangles
Instead of triangles, we can generalize the previous analysis to forbidcopies of K`+1, ` ≥ 2.
Theorem (Turan, 1941)
If an n-vertex graph H has no copy of K`, then the number of edges H hasis at most
n2
2`.
This bound is only achieved if H is complete `-partite and ` | n.
So, the maximum number of changes required to remove triangles fromn-vertex graph G is
∼ n2
2`
∼ 1
`
(n
2
).
This is achieved by G = Kn.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 5 / 19
![Page 14: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/14.jpg)
Results on triangles
Instead of triangles, we can generalize the previous analysis to forbidcopies of K`+1, ` ≥ 2.
Theorem (Turan, 1941)
If an n-vertex graph H has no copy of K`, then the number of edges H hasis at most
n2
2`.
This bound is only achieved if H is complete `-partite and ` | n.
So, the maximum number of changes required to remove triangles fromn-vertex graph G is
∼ n2
2`
∼ 1
`
(n
2
).
This is achieved by G = Kn.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 5 / 19
![Page 15: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/15.jpg)
Results on triangles
Instead of triangles, we can generalize the previous analysis to forbidcopies of K`+1, ` ≥ 2.
Theorem (Turan, 1941)
If an n-vertex graph H has no copy of K`, then the number of edges H hasis at most
n2
2`.
This bound is only achieved if H is complete `-partite and ` | n.
So, the maximum number of changes required to remove triangles fromn-vertex graph G is
∼ n2
2`∼ 1
`
(n
2
).
This is achieved by G = Kn.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 5 / 19
![Page 16: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/16.jpg)
Editing a bipartite graph
The edit distance question began with the following question of Chen,Eulenstein, Fernandez-Baca and Sanderson:
Given A bipartite graph, G = (A,B; E ), |A| = |B| = N.
An induced “” (edges in red, nonedges in blue):
The question relates to consensus trees. Two trees are comparable if acorresponding bipartite graph has no induced W or M.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 6 / 19
![Page 17: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/17.jpg)
Editing a bipartite graph
The edit distance question began with the following question of Chen,Eulenstein, Fernandez-Baca and Sanderson:
Given A bipartite graph, G = (A,B; E ), |A| = |B| = N.
An induced “” (edges in red, nonedges in blue):
The question relates to consensus trees. Two trees are comparable if acorresponding bipartite graph has no induced W or M.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 6 / 19
![Page 18: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/18.jpg)
Editing a bipartite graph
The edit distance question began with the following question of Chen,Eulenstein, Fernandez-Baca and Sanderson:
Given A bipartite graph, G = (A,B; E ), |A| = |B| = N.
Question
How many edge-deletions plus edge-additions are necessary to ensure thatG has no copy of “W ” as an induced subgraph?
An induced “W ” (edges in red, nonedges in blue):
The question relates to consensus trees. Two trees are comparable if acorresponding bipartite graph has no induced W or M.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 6 / 19
![Page 19: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/19.jpg)
Editing a bipartite graph
The edit distance question began with the following question of Chen,Eulenstein, Fernandez-Baca and Sanderson:
Given A bipartite graph, G = (A,B; E ), |A| = |B| = N.
Question
How many edge-deletions plus edge-additions are necessary to ensure thatG has no copy of “M” as an induced subgraph?
An induced “M” (edges in red, nonedges in blue):
The question relates to consensus trees. Two trees are comparable if acorresponding bipartite graph has no induced W or M.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 6 / 19
![Page 20: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/20.jpg)
Editing a bipartite graph
The edit distance question began with the following question of Chen,Eulenstein, Fernandez-Baca and Sanderson:
Given A bipartite graph, G = (A,B; E ), |A| = |B| = N.
Question
How many edge-deletions plus edge-additions are necessary to ensure thatG has no copy of “W or M” as an induced subgraph?
An induced “” (edges in red, nonedges in blue):
The question relates to consensus trees. Two trees are comparable if acorresponding bipartite graph has no induced W or M.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 6 / 19
![Page 21: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/21.jpg)
Bipartite graph editing settled asymptotically
Szemeredi’s Regularity Lemma gives:
Theorem (Axenovich-M. 2006)
Let H be any fixed bipartite graph that is neither empty nor complete.The number of edge-operations necessary to remove all induced copies ofH from a random N ×N bipartite graph is at least (1/2)N2 − o(N2), withhigh probability.
Clearly, (1/2)N2 edge operations suffices.
The more general case is not so easy to settle, even asymptotically.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 7 / 19
![Page 22: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/22.jpg)
Bipartite graph editing settled asymptotically
Szemeredi’s Regularity Lemma gives:
Theorem (Axenovich-M. 2006)
Let H be any fixed bipartite graph that is neither empty nor complete.The number of edge-operations necessary to remove all induced copies ofH from a random N ×N bipartite graph is at least (1/2)N2 − o(N2), withhigh probability.
Clearly, (1/2)N2 edge operations suffices.
The more general case is not so easy to settle, even asymptotically.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 7 / 19
![Page 23: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/23.jpg)
Edit Distance between graphs
Given: Labeled graphs G and G ′, each on n vertices.
Definition
The edit distance between G and G ′
Dist(G ,G ′) =∣∣E (G )4E (G ′)
∣∣is the size of the symmetric difference of the edge sets.
That is, it is the minimum number of edge-additions plus edge-deletions totransform G into G ′.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 8 / 19
![Page 24: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/24.jpg)
Edit Distance between graphs
Given: Labeled graphs G and G ′, each on n vertices.
Definition
The edit distance between G and G ′
Dist(G ,G ′) =∣∣E (G )4E (G ′)
∣∣is the size of the symmetric difference of the edge sets.
That is, it is the minimum number of edge-additions plus edge-deletions totransform G into G ′.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 8 / 19
![Page 25: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/25.jpg)
Edit Distance between graphs
Given: Labeled graphs G and G ′, each on n vertices.
Definition
The edit distance between G and G ′
Dist(G ,G ′) =∣∣E (G )4E (G ′)
∣∣is the size of the symmetric difference of the edge sets.
That is, it is the minimum number of edge-additions plus edge-deletions totransform G into G ′.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 8 / 19
![Page 26: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/26.jpg)
Edit Distance from properties
Given: A labeled graph G and a graph property P.A graph property is a set of graphs.
Definition
The edit distance from G to P
Dist(G ,P) = min{Dist(G ,G ′) : G ′ ∈ P
}is the least edit distance of G to a graph in P.
That is, it is the minimum number of edge-additions plus edge-deletions totransform G into a member of P.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 9 / 19
![Page 27: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/27.jpg)
Edit Distance from properties
Given: A labeled graph G and a graph property P.A graph property is a set of graphs.
Definition
The edit distance from G to P
Dist(G ,P) = min{Dist(G ,G ′) : G ′ ∈ P
}is the least edit distance of G to a graph in P.
That is, it is the minimum number of edge-additions plus edge-deletions totransform G into a member of P.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 9 / 19
![Page 28: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/28.jpg)
Edit Distance from properties
Given: A labeled graph G and a graph property P.A graph property is a set of graphs.
Definition
The edit distance from G to P
Dist(G ,P) = min{Dist(G ,G ′) : G ′ ∈ P
}is the least edit distance of G to a graph in P.
That is, it is the minimum number of edge-additions plus edge-deletions totransform G into a member of P.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 9 / 19
![Page 29: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/29.jpg)
Extremal Edit Distance
Given: A natural number n and a graph property P.
Definition
The edit distance from P
Dist(n,P) = max {Dist(G ,P) : |V (G )| = n}
is the maximum edit distance of an n-vertex graph to a graph in P.
That is, it is the maximum, over all n-vertex graphs, G , of the minimumnumber of edge-additions plus edge-deletions to transform G into amember of P.
A hereditary property is one that is preserved undervertex-deletion. Example: Forb(K3,3), no induced copy of K3,3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 10 / 19
![Page 30: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/30.jpg)
Extremal Edit Distance
Given: A natural number n and a graph property P.
Definition
The edit distance from P
Dist(n,P) = max {Dist(G ,P) : |V (G )| = n}
is the maximum edit distance of an n-vertex graph to a graph in P.
That is, it is the maximum, over all n-vertex graphs, G , of the minimumnumber of edge-additions plus edge-deletions to transform G into amember of P.
A hereditary property is one that is preserved undervertex-deletion. Example: Forb(K3,3), no induced copy of K3,3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 10 / 19
![Page 31: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/31.jpg)
Extremal Edit Distance
Given: A natural number n and a graph property P.
Definition
The edit distance from P
Dist(n,P) = max {Dist(G ,P) : |V (G )| = n}
is the maximum edit distance of an n-vertex graph to a graph in P.
That is, it is the maximum, over all n-vertex graphs, G , of the minimumnumber of edge-additions plus edge-deletions to transform G into amember of P.
A hereditary property is one that is preserved undervertex-deletion. Example: Forb(K3,3), no induced copy of K3,3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 10 / 19
![Page 32: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/32.jpg)
Extremal Edit Distance
Given: A natural number n and a hereditary graph property H.
Definition
The edit distance from H
Dist(n,H) = max {Dist(G ,H) : |V (G )| = n}
is the maximum edit distance of an n-vertex graph to a graph in H.
That is, it is the maximum, over all n-vertex graphs, G , of the minimumnumber of edge-additions plus edge-deletions to transform G into amember of H.
A hereditary property is one that is preserved undervertex-deletion.
Example: Forb(K3,3), no induced copy of K3,3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 10 / 19
![Page 33: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/33.jpg)
Extremal Edit Distance
Given: A natural number n and a hereditary graph property H.
Definition
The edit distance from H
Dist(n,H) = max {Dist(G ,H) : |V (G )| = n}
is the maximum edit distance of an n-vertex graph to a graph in H.
That is, it is the maximum, over all n-vertex graphs, G , of the minimumnumber of edge-additions plus edge-deletions to transform G into amember of H.
A hereditary property is one that is preserved undervertex-deletion. Example: Forb(K3,3), no induced copy of K3,3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 10 / 19
![Page 34: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/34.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
Examples:
Planar graphs.
Line graphs of bipartite graphs.
Chordal graphs
: graphs with no chordless cycle of length ≥ 4.
Perfect graphs
: χ = ω for all induced subgraphs.
Forb(H)
: graphs with no induced copy of H.
(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 35: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/35.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
I.e., if G ∈ H, then every induced subgraph of G is in H.
Examples:
Planar graphs.Line graphs of bipartite graphs.Chordal graphs
: graphs with no chordless cycle of length ≥ 4.
Perfect graphs
: χ = ω for all induced subgraphs.
Forb(H)
: graphs with no induced copy of H.
(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 36: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/36.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
I.e., if G ∈ H, then every induced subgraph of G is in H.
A 5-cycle as a subgraph, but no induced 5-cycle.
Examples:
Planar graphs.Line graphs of bipartite graphs.Chordal graphs
: graphs with no chordless cycle of length ≥ 4.
Perfect graphs
: χ = ω for all induced subgraphs.
Forb(H)
: graphs with no induced copy of H.
(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 37: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/37.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
I.e., if G ∈ H, then every induced subgraph of G is in H.
Examples:
Planar graphs.
Line graphs of bipartite graphs.Chordal graphs
: graphs with no chordless cycle of length ≥ 4.
Perfect graphs
: χ = ω for all induced subgraphs.
Forb(H)
: graphs with no induced copy of H.
(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 38: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/38.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
I.e., if G ∈ H, then every induced subgraph of G is in H.
Examples:
Planar graphs.Line graphs of bipartite graphs.
Chordal graphs
: graphs with no chordless cycle of length ≥ 4.
Perfect graphs
: χ = ω for all induced subgraphs.
Forb(H)
: graphs with no induced copy of H.
(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 39: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/39.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
I.e., if G ∈ H, then every induced subgraph of G is in H.
Examples:
Planar graphs.Line graphs of bipartite graphs.Chordal graphs
: graphs with no chordless cycle of length ≥ 4.Perfect graphs
: χ = ω for all induced subgraphs.
Forb(H)
: graphs with no induced copy of H.
(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 40: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/40.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
I.e., if G ∈ H, then every induced subgraph of G is in H.
Examples:
Planar graphs.Line graphs of bipartite graphs.Chordal graphs: graphs with no chordless cycle of length ≥ 4.
Perfect graphs
: χ = ω for all induced subgraphs.
Forb(H)
: graphs with no induced copy of H.
(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 41: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/41.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
I.e., if G ∈ H, then every induced subgraph of G is in H.
Examples:
Planar graphs.Line graphs of bipartite graphs.Chordal graphs: graphs with no chordless cycle of length ≥ 4.Perfect graphs
: χ = ω for all induced subgraphs.Forb(H)
: graphs with no induced copy of H.
(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 42: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/42.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
I.e., if G ∈ H, then every induced subgraph of G is in H.
Examples:
Planar graphs.Line graphs of bipartite graphs.Chordal graphs: graphs with no chordless cycle of length ≥ 4.Perfect graphs: χ = ω for all induced subgraphs.
Forb(H)
: graphs with no induced copy of H.
(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 43: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/43.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
I.e., if G ∈ H, then every induced subgraph of G is in H.
Examples:
Planar graphs.Line graphs of bipartite graphs.Chordal graphs: graphs with no chordless cycle of length ≥ 4.Perfect graphs: χ = ω for all induced subgraphs.Forb(H)
: graphs with no induced copy of H.
(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 44: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/44.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
I.e., if G ∈ H, then every induced subgraph of G is in H.
Examples:
Planar graphs.Line graphs of bipartite graphs.Chordal graphs: graphs with no chordless cycle of length ≥ 4.Perfect graphs: χ = ω for all induced subgraphs.Forb(H): graphs with no induced copy of H.
(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 45: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/45.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
I.e., if G ∈ H, then every induced subgraph of G is in H.
Examples:
Planar graphs.Line graphs of bipartite graphs.Chordal graphs: graphs with no chordless cycle of length ≥ 4.Perfect graphs: χ = ω for all induced subgraphs.Forb(H): graphs with no induced copy of H.(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 46: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/46.jpg)
Hereditary Properties
Definition
A hereditary property, H, of graphs is one that holds under thedeletion of vertices.
Examples:
Planar graphs.
Line graphs of bipartite graphs.
Chordal graphs: graphs with no chordless cycle of length ≥ 4.
Perfect graphs: χ = ω for all induced subgraphs.
Forb(H): graphs with no induced copy of H.(Forb(H) is a principal hereditary property.)
For the rest of this talk, all of our hereditary properties are principal; i.e.,
H = Forb(H), for some graph H.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 11 / 19
![Page 47: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/47.jpg)
A useful parameter
Let H = Forb(H) and let a, c ∈ N have the property that V (H) cannot bepartitioned into a set of a independent sets and c cliques.
But V (H) can be partitioned into any combination of a + c + 1independent sets and cliques.
We call a + c + 1 the binary chromatic number, χB(H).
Theorem (Axenovich-Kezdy-M., 2008)
Let H = Forb(H) for some fixed graph H which has binary chromaticnumber χB(H),
Dist(n,Forb(H)) ≥ 1
2(χB(H)− 1)
(n
2
)− o(n2).
Moreover, this holds with equality if H is self-complementary.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 12 / 19
![Page 48: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/48.jpg)
A useful parameter
Let H = Forb(H) and let a, c ∈ N have the property that V (H) cannot bepartitioned into a set of a independent sets and c cliques.
But V (H) can be partitioned into any combination of a + c + 1independent sets and cliques.
We call a + c + 1 the binary chromatic number, χB(H).
Theorem (Axenovich-Kezdy-M., 2008)
Let H = Forb(H) for some fixed graph H which has binary chromaticnumber χB(H),
Dist(n,Forb(H)) ≥ 1
2(χB(H)− 1)
(n
2
)− o(n2).
Moreover, this holds with equality if H is self-complementary.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 12 / 19
![Page 49: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/49.jpg)
A useful parameter
Let H = Forb(H) and let a, c ∈ N have the property that V (H) cannot bepartitioned into a set of a independent sets and c cliques.
But V (H) can be partitioned into any combination of a + c + 1independent sets and cliques.
We call a + c + 1 the binary chromatic number, χB(H).
Theorem (Axenovich-Kezdy-M., 2008)
Let H = Forb(H) for some fixed graph H which has binary chromaticnumber χB(H),
Dist(n,Forb(H)) ≥ 1
2(χB(H)− 1)
(n
2
)− o(n2).
Moreover, this holds with equality if H is self-complementary.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 12 / 19
![Page 50: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/50.jpg)
A useful parameter
Let H = Forb(H) and let a, c ∈ N have the property that V (H) cannot bepartitioned into a set of a independent sets and c cliques.
But V (H) can be partitioned into any combination of a + c + 1independent sets and cliques.
We call a + c + 1 the binary chromatic number, χB(H).
Theorem (Axenovich-Kezdy-M., 2008)
Let H = Forb(H) for some fixed graph H which has binary chromaticnumber χB(H),
Dist(n,Forb(H)) ≥ 1
2(χB(H)− 1)
(n
2
)− o(n2).
Moreover, this holds with equality if H is self-complementary.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 12 / 19
![Page 51: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/51.jpg)
An example: 5-cycle
We will compute χB for the 5-cycle, C5.
So, χB(C5)3.
But it cannot be partitioned into, say, 2 independent sets and 0 cliques.
So, χB(C5)3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 13 / 19
![Page 52: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/52.jpg)
An example: 5-cycle
We will compute χB for the 5-cycle, C5.
Let us consider all (a, c) such that a + c + 1 = 3:
The 5-cycle can be partitioned into3 independent sets and 0 cliques.
So, χB(C5)3.
But it cannot be partitioned into, say, 2 independent sets and 0 cliques.
So, χB(C5)3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 13 / 19
![Page 53: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/53.jpg)
An example: 5-cycle
We will compute χB for the 5-cycle, C5.
Let us consider all (a, c) such that a + c + 1 = 3:
The 5-cycle can be partitioned into2 independent sets and 1 cliques.
So, χB(C5)3.
But it cannot be partitioned into, say, 2 independent sets and 0 cliques.
So, χB(C5)3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 13 / 19
![Page 54: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/54.jpg)
An example: 5-cycle
We will compute χB for the 5-cycle, C5.
Let us consider all (a, c) such that a + c + 1 = 3:
The 5-cycle can be partitioned into1 independent sets and 2 cliques.
So, χB(C5)3.
But it cannot be partitioned into, say, 2 independent sets and 0 cliques.
So, χB(C5)3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 13 / 19
![Page 55: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/55.jpg)
An example: 5-cycle
We will compute χB for the 5-cycle, C5.
Let us consider all (a, c) such that a + c + 1 = 3:
The 5-cycle can be partitioned into0 independent sets and 3 cliques.
So, χB(C5)3.
But it cannot be partitioned into, say, 2 independent sets and 0 cliques.
So, χB(C5)3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 13 / 19
![Page 56: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/56.jpg)
An example: 5-cycle
We will compute χB for the 5-cycle, C5.
Let us consider all (a, c) such that a + c + 1 = 3:
The 5-cycle can be partitioned into0 independent sets and 3 cliques.
So, χB(C5) ≤ 3.
But it cannot be partitioned into, say, 2 independent sets and 0 cliques.
So, χB(C5)3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 13 / 19
![Page 57: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/57.jpg)
An example: 5-cycle
We will compute χB for the 5-cycle, C5.
So, χB(C5) ≤ 3.
But it cannot be partitioned into, say, 2 independent sets and 0 cliques.
So, χB(C5)3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 13 / 19
![Page 58: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/58.jpg)
An example: 5-cycle
We will compute χB for the 5-cycle, C5.
So, χB(C5) ≤ 3.
But it cannot be partitioned into, say, 2 independent sets and 0 cliques.
So, χB(C5) ≥ 3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 13 / 19
![Page 59: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/59.jpg)
An example: 5-cycle
We will compute χB for the 5-cycle, C5.
So, χB(C5) = 3.
But it cannot be partitioned into, say, 2 independent sets and 0 cliques.
So, χB(C5) = 3.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 13 / 19
![Page 60: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/60.jpg)
Edit distance for C5
Since χB(C5) = 3, and C5 is self-complementary, the theorem gives
Dist(n,Forb(H)) =1
2(χB(H)− 1)
(n
2
)− o(n2).
Normalize and take the limit:
limn→∞
Dist(n,Forb(H))
(n
2
)−1
=1
4
− o(1)
.
We denote
d∗(H)def= lim
n→∞Dist(n,H)
(n
2
)−1
.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 14 / 19
![Page 61: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/61.jpg)
Edit distance for C5
Since χB(C5) = 3, and C5 is self-complementary, the theorem gives
Dist(n,Forb(H)) =1
2(3− 1)
(n
2
)− o(n2).
Normalize and take the limit:
limn→∞
Dist(n,Forb(H))
(n
2
)−1
=1
4
− o(1)
.
We denote
d∗(H)def= lim
n→∞Dist(n,H)
(n
2
)−1
.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 14 / 19
![Page 62: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/62.jpg)
Edit distance for C5
Since χB(C5) = 3, and C5 is self-complementary, the theorem gives
Dist(n,Forb(H)) =1
4
(n
2
)− o(n2).
Normalize and take the limit:
limn→∞
Dist(n,Forb(H))
(n
2
)−1
=1
4
− o(1)
.
We denote
d∗(H)def= lim
n→∞Dist(n,H)
(n
2
)−1
.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 14 / 19
![Page 63: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/63.jpg)
Edit distance for C5
Since χB(C5) = 3, and C5 is self-complementary, the theorem gives
Dist(n,Forb(H)) =1
4
(n
2
)− o(n2).
Normalize and take the limit:
limn→∞
Dist(n,Forb(H))
(n
2
)−1
=1
4− o(1).
We denote
d∗(H)def= lim
n→∞Dist(n,H)
(n
2
)−1
.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 14 / 19
![Page 64: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/64.jpg)
Edit distance for C5
Since χB(C5) = 3, and C5 is self-complementary, the theorem gives
Dist(n,Forb(H)) =1
4
(n
2
)− o(n2).
Normalize and take the limit:
limn→∞
Dist(n,Forb(H))
(n
2
)−1
=1
4
− o(1)
.
We denote
d∗(H)def= lim
n→∞Dist(n,H)
(n
2
)−1
.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 14 / 19
![Page 65: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/65.jpg)
Understanding d∗
Let Gn,p denote the Erdos-Renyi random graph:
I.e., there are n vertices and each edge is present, independently, withprobability p.
Theorem (Alon-Stav, 2008)
For every hereditary property, H, there exists a p∗ = p∗(H) ∈ [0, 1] suchthat
d∗(H) = limn→∞
Dist (Gn,p∗ ,H)
(n
2
)−1
.
Theorem (Balogh-M., 2008)
For every hereditary property, H, and every p ∈ [0, 1], if
gH(p) = limn→∞
Dist (Gn,p,H)
(n
2
)−1
then it is also true that
gH(p) = limn→∞
max
{Dist (G ,H) : |V (G )| = n, |E (G )| = p
(n
2
)}(n
2
)−1
.
Roughly, the hardest density-p graph to edit is the random graph.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 15 / 19
![Page 66: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/66.jpg)
Understanding d∗
Let Gn,p denote the Erdos-Renyi random graph:
I.e., there are n vertices and each edge is present, independently, withprobability p.
Theorem (Alon-Stav, 2008)
For every hereditary property, H, there exists a p∗ = p∗(H) ∈ [0, 1] suchthat
d∗(H) = limn→∞
Dist (Gn,p∗ ,H)
(n
2
)−1
.
(The expression Dist (Gn,p∗ ,H) is tightly concentrated about the mean, sothe right-hand side is well-defined.)
Theorem (Balogh-M., 2008)
For every hereditary property, H, and every p ∈ [0, 1], if
gH(p) = limn→∞
Dist (Gn,p,H)
(n
2
)−1
then it is also true that
gH(p) = limn→∞
max
{Dist (G ,H) : |V (G )| = n, |E (G )| = p
(n
2
)}(n
2
)−1
.
Roughly, the hardest density-p graph to edit is the random graph.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 15 / 19
![Page 67: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/67.jpg)
Understanding d∗
Theorem (Alon-Stav, 2008)
For every hereditary property, H, there exists a p∗ = p∗(H) ∈ [0, 1] suchthat
d∗(H) = limn→∞
Dist (Gn,p∗ ,H)
(n
2
)−1
.
Theorem (Balogh-M., 2008)
For every hereditary property, H, and every p ∈ [0, 1], if
gH(p) = limn→∞
Dist (Gn,p,H)
(n
2
)−1
then it is also true that
gH(p) = limn→∞
max
{Dist (G ,H) : |V (G )| = n, |E (G )| = p
(n
2
)}(n
2
)−1
.
Roughly, the hardest density-p graph to edit is the random graph.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 15 / 19
![Page 68: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/68.jpg)
Understanding d∗
Theorem (Balogh-M., 2008)
For every hereditary property, H, and every p ∈ [0, 1], if
gH(p) = limn→∞
Dist (Gn,p,H)
(n
2
)−1
then it is also true that
gH(p) = limn→∞
max
{Dist (G ,H) : |V (G )| = n, |E (G )| = p
(n
2
)}(n
2
)−1
.
Roughly, the hardest density-p graph to edit is the random graph.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 15 / 19
![Page 69: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/69.jpg)
The Edit Distance Function
Properties of gForb(H)(p)
Continuous and concave down.
Achieves its maximum (p∗, d∗) for some p∗ ∈ [0, 1].
gH(p)
(1
2
)=
1
2 (χB(H)− 1).
If H is neither complete nor empty, then gH(0) = gHH(1) = 0.
For any rational r ∈ [0, 1], there is an H, such that p∗(Forb(H)) = r .
There is an irrational q ∈ [0, 1] and an H, such that p∗(Forb(H)) = q.
Theorem (Balogh-M., 2008)
p∗(Forb(K3,3)) =√
2− 1 d∗(Forb(K3,3)) = 3− 2√
2.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 16 / 19
![Page 70: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/70.jpg)
The Edit Distance Function
Properties of gForb(H)(p)
Continuous and concave down.
Achieves its maximum (p∗, d∗) for some p∗ ∈ [0, 1].
gH(p)
(1
2
)=
1
2 (χB(H)− 1).
If H is neither complete nor empty, then gH(0) = gHH(1) = 0.
For any rational r ∈ [0, 1], there is an H, such that p∗(Forb(H)) = r .
There is an irrational q ∈ [0, 1] and an H, such that p∗(Forb(H)) = q.
Theorem (Balogh-M., 2008)
p∗(Forb(K3,3)) =√
2− 1 d∗(Forb(K3,3)) = 3− 2√
2.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 16 / 19
![Page 71: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/71.jpg)
The Edit Distance Function
Properties of gForb(H)(p)
Continuous and concave down.
Achieves its maximum (p∗, d∗) for some p∗ ∈ [0, 1].
gH(p)
(1
2
)=
1
2 (χB(H)− 1).
If H is neither complete nor empty, then gH(0) = gHH(1) = 0.
For any rational r ∈ [0, 1], there is an H, such that p∗(Forb(H)) = r .
There is an irrational q ∈ [0, 1] and an H, such that p∗(Forb(H)) = q.
Theorem (Balogh-M., 2008)
p∗(Forb(K3,3)) =√
2− 1 d∗(Forb(K3,3)) = 3− 2√
2.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 16 / 19
![Page 72: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/72.jpg)
The Edit Distance Function
Properties of gForb(H)(p)
Continuous and concave down.
Achieves its maximum (p∗, d∗) for some p∗ ∈ [0, 1].
gH(p)
(1
2
)=
1
2 (χB(H)− 1).
If H is neither complete nor empty, then gH(0) = gHH(1) = 0.
For any rational r ∈ [0, 1], there is an H, such that p∗(Forb(H)) = r .
There is an irrational q ∈ [0, 1] and an H, such that p∗(Forb(H)) = q.
Theorem (Balogh-M., 2008)
p∗(Forb(K3,3)) =√
2− 1 d∗(Forb(K3,3)) = 3− 2√
2.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 16 / 19
![Page 73: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/73.jpg)
The Edit Distance Function
Properties of gForb(H)(p)
Continuous and concave down.
Achieves its maximum (p∗, d∗) for some p∗ ∈ [0, 1].
gH(p)
(1
2
)=
1
2 (χB(H)− 1).
If H is neither complete nor empty, then gH(0) = gHH(1) = 0.
For any rational r ∈ [0, 1], there is an H, such that p∗(Forb(H)) = r .
There is an irrational q ∈ [0, 1] and an H, such that p∗(Forb(H)) = q.
Theorem (Balogh-M., 2008)
p∗(Forb(K3,3)) =√
2− 1 d∗(Forb(K3,3)) = 3− 2√
2.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 16 / 19
![Page 74: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/74.jpg)
The Edit Distance Function
Properties of gForb(H)(p)
Continuous and concave down.
Achieves its maximum (p∗, d∗) for some p∗ ∈ [0, 1].
gH(p)
(1
2
)=
1
2 (χB(H)− 1).
If H is neither complete nor empty, then gH(0) = gHH(1) = 0.
For any rational r ∈ [0, 1], there is an H, such that p∗(Forb(H)) = r .
There is an irrational q ∈ [0, 1] and an H, such that p∗(Forb(H)) = q.
Theorem (Balogh-M., 2008)
p∗(Forb(Ka + Eb)) =a− 1
a + b − 1d∗(Forb(Ka + Eb)) =
1
a + b − 1.
Theorem (Balogh-M., 2008)
p∗(Forb(K3,3)) =√
2− 1 d∗(Forb(K3,3)) = 3− 2√
2.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 16 / 19
![Page 75: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/75.jpg)
The Edit Distance Function
Properties of gForb(H)(p)
Continuous and concave down.
Achieves its maximum (p∗, d∗) for some p∗ ∈ [0, 1].
gH(p)
(1
2
)=
1
2 (χB(H)− 1).
If H is neither complete nor empty, then gH(0) = gHH(1) = 0.
For any rational r ∈ [0, 1], there is an H, such that p∗(Forb(H)) = r .
There is an irrational q ∈ [0, 1] and an H, such that p∗(Forb(H)) = q.
Theorem (Balogh-M., 2008)
p∗(Forb(K3,3)) =√
2− 1 d∗(Forb(K3,3)) = 3− 2√
2.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 16 / 19
![Page 76: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/76.jpg)
Weighted Turan lemma
The value of computing the edit distance for specific hereditary propertiesare the techniques used and how they might be generalized.
Lemma (Balogh-M., 2008)
Let a ≥ 2 and K , |V (K )| = k be a graph with edges colored BLACK,WHITE and GRAY, with the property that any set A of a vertices has atleast one of the following conditions:
1 A contains at least one WHITE edge;
2 A contains a spanning subgraph of BLACK edges.
With EW(K ) denoting the white edges and EB(K ) the black edges,
(a− 1)|EW(K )|+ |EB(K )| ≥⌈
k
2(k − a + 1)
⌉.
If we only apply (1) and not (2), then, with a fixed, Turan’s theorem is,asymptotically, a consequence.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 17 / 19
![Page 77: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/77.jpg)
Weighted Turan lemma
The value of computing the edit distance for specific hereditary propertiesare the techniques used and how they might be generalized.
Lemma (Balogh-M., 2008)
Let a ≥ 2 and K , |V (K )| = k be a graph with edges colored BLACK,WHITE and GRAY, with the property that any set A of a vertices has atleast one of the following conditions:
1 A contains at least one WHITE edge;
2 A contains a spanning subgraph of BLACK edges.
With EW(K ) denoting the white edges and EB(K ) the black edges,
(a− 1)|EW(K )|+ |EB(K )| ≥⌈
k
2(k − a + 1)
⌉.
If we only apply (1) and not (2), then, with a fixed, Turan’s theorem is,asymptotically, a consequence.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 17 / 19
![Page 78: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/78.jpg)
Weighted Turan lemma
The value of computing the edit distance for specific hereditary propertiesare the techniques used and how they might be generalized.
Lemma (Balogh-M., 2008)
Let a ≥ 2 and K , |V (K )| = k be a graph with edges colored BLACK,WHITE and GRAY, with the property that any set A of a vertices has atleast one of the following conditions:
1 A contains at least one WHITE edge;
2 A contains a spanning subgraph of BLACK edges.
With EW(K ) denoting the white edges and EB(K ) the black edges,
(a− 1)|EW(K )|+ |EB(K )| ≥⌈
k
2(k − a + 1)
⌉.
If we only apply (1) and not (2), then, with a fixed, Turan’s theorem is,asymptotically, a consequence.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 17 / 19
![Page 79: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/79.jpg)
Future Work
gH(p) is a function of H and is derived from the random graphs Gn,p.What other functions of H have similar properties? The followingfunction has been studied and shares some of the properties of gH(p):
− log2 Pr(Gn,p ∈ H).
For which metrics on graph spaces do similar properties hold to the(normalized) edit metric? The metric derived from the so-called cutnorm, for example, is well-studied and shares some properties withedit.
What are the axioms that will define such metrics as being “good”?
Graph limits are new but well-studied. The limits of graphs arefunctions of a measure space on [0, 1]2. Balazs Szegedy reports thatlimits of hereditary properties can be identified. Graph limits andgraph metrics may share a common theoretical bond.
Conjecture
Fix p0 ∈ [0, 1] and let H ∼ G (n0, p0) with H = Forb(H). Then,
gH(p) ∼ 2 log2 n0
n0min
{p
log21
1−p0
,1− p
log21p0
}.
with prob. → 1 as n0 →∞.
Note that this would imply the counterintuitive(?)
p∗ ∼ log(1− p0)
log(p0(1− p0)).
Counterintuitive because log(1−p0)log(p0(1−p0))
= p0 if and only if p0 ∈ {0, 1/2, 1}.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 18 / 19
![Page 80: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/80.jpg)
Future Work
gH(p) is a function of H and is derived from the random graphs Gn,p.What other functions of H have similar properties? The followingfunction has been studied and shares some of the properties of gH(p):
− log2 Pr(Gn,p ∈ H).
For which metrics on graph spaces do similar properties hold to the(normalized) edit metric? The metric derived from the so-called cutnorm, for example, is well-studied and shares some properties withedit.
What are the axioms that will define such metrics as being “good”?
Graph limits are new but well-studied. The limits of graphs arefunctions of a measure space on [0, 1]2. Balazs Szegedy reports thatlimits of hereditary properties can be identified. Graph limits andgraph metrics may share a common theoretical bond.
Conjecture
Fix p0 ∈ [0, 1] and let H ∼ G (n0, p0) with H = Forb(H). Then,
gH(p) ∼ 2 log2 n0
n0min
{p
log21
1−p0
,1− p
log21p0
}.
with prob. → 1 as n0 →∞.
Note that this would imply the counterintuitive(?)
p∗ ∼ log(1− p0)
log(p0(1− p0)).
Counterintuitive because log(1−p0)log(p0(1−p0))
= p0 if and only if p0 ∈ {0, 1/2, 1}.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 18 / 19
![Page 81: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/81.jpg)
Future Work
gH(p) is a function of H and is derived from the random graphs Gn,p.What other functions of H have similar properties? The followingfunction has been studied and shares some of the properties of gH(p):
− log2 Pr(Gn,p ∈ H).
For which metrics on graph spaces do similar properties hold to the(normalized) edit metric? The metric derived from the so-called cutnorm, for example, is well-studied and shares some properties withedit.What are the axioms that will define such metrics as being “good”?
Graph limits are new but well-studied. The limits of graphs arefunctions of a measure space on [0, 1]2. Balazs Szegedy reports thatlimits of hereditary properties can be identified. Graph limits andgraph metrics may share a common theoretical bond.
Conjecture
Fix p0 ∈ [0, 1] and let H ∼ G (n0, p0) with H = Forb(H). Then,
gH(p) ∼ 2 log2 n0
n0min
{p
log21
1−p0
,1− p
log21p0
}.
with prob. → 1 as n0 →∞.
Note that this would imply the counterintuitive(?)
p∗ ∼ log(1− p0)
log(p0(1− p0)).
Counterintuitive because log(1−p0)log(p0(1−p0))
= p0 if and only if p0 ∈ {0, 1/2, 1}.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 18 / 19
![Page 82: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/82.jpg)
Future Work
gH(p) is a function of H and is derived from the random graphs Gn,p.What other functions of H have similar properties? The followingfunction has been studied and shares some of the properties of gH(p):
− log2 Pr(Gn,p ∈ H).
For which metrics on graph spaces do similar properties hold to the(normalized) edit metric? The metric derived from the so-called cutnorm, for example, is well-studied and shares some properties withedit.What are the axioms that will define such metrics as being “good”?
Graph limits are new but well-studied. The limits of graphs arefunctions of a measure space on [0, 1]2. Balazs Szegedy reports thatlimits of hereditary properties can be identified. Graph limits andgraph metrics may share a common theoretical bond.
Conjecture
Fix p0 ∈ [0, 1] and let H ∼ G (n0, p0) with H = Forb(H). Then,
gH(p) ∼ 2 log2 n0
n0min
{p
log21
1−p0
,1− p
log21p0
}.
with prob. → 1 as n0 →∞.
Note that this would imply the counterintuitive(?)
p∗ ∼ log(1− p0)
log(p0(1− p0)).
Counterintuitive because log(1−p0)log(p0(1−p0))
= p0 if and only if p0 ∈ {0, 1/2, 1}.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 18 / 19
![Page 83: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/83.jpg)
Future Work
Conjecture
Fix p0 ∈ [0, 1] and let H ∼ G (n0, p0) with H = Forb(H). Then,
gH(p) ∼ 2 log2 n0
n0min
{p
log21
1−p0
,1− p
log21p0
}.
with prob. → 1 as n0 →∞.
Note that this would imply the counterintuitive(?)
p∗ ∼ log(1− p0)
log(p0(1− p0)).
Counterintuitive because log(1−p0)log(p0(1−p0))
= p0 if and only if p0 ∈ {0, 1/2, 1}.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 18 / 19
![Page 84: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/84.jpg)
Future Work
Conjecture
Fix p0 ∈ [0, 1] and let H ∼ G (n0, p0) with H = Forb(H). Then,
gH(p) ∼ 2 log2 n0
n0min
{p
log21
1−p0
,1− p
log21p0
}.
with prob. → 1 as n0 →∞.
Note that this would imply the counterintuitive(?)
p∗ ∼ log(1− p0)
log(p0(1− p0)).
Counterintuitive because log(1−p0)log(p0(1−p0))
= p0 if and only if p0 ∈ {0, 1/2, 1}.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 18 / 19
![Page 85: 1 Extremal graph theory - Iowa State University · 2008-10-27 · 1 Extremal graph theory 2 Edit Distance Origins 3 De nitions related to Edit Distance 4 Hereditary Properties 5 Binary](https://reader030.vdocuments.us/reader030/viewer/2022041013/5ec42bcd167ddc3c103f88a4/html5/thumbnails/85.jpg)
Future Work
Conjecture
Fix p0 ∈ [0, 1] and let H ∼ G (n0, p0) with H = Forb(H). Then,
gH(p) ∼ 2 log2 n0
n0min
{p
log21
1−p0
,1− p
log21p0
}.
with prob. → 1 as n0 →∞.
Note that this would imply the counterintuitive(?)
p∗ ∼ log(1− p0)
log(p0(1− p0)).
Counterintuitive because log(1−p0)log(p0(1−p0))
= p0 if and only if p0 ∈ {0, 1/2, 1}.
Ryan Martin (Iowa State U.) The edit distance in graphs 27 October 2008 18 / 19