Transcript Document

Reconstruction Algorithm
for Permutation Graphs
Masashi Kiyomi, Toshiki Saitoh,
and Ryuhei Uehara
School of Information Science
Japan Advanced Institute of Science and Technology
Graph Reconstruction Problem
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
Deck of Graph G=(V, E): multi-set {G-v | v∈V}
Preimage of multi-set D: a graph whose deck is D
deck of G
v1
v4
v1
v2
v3
v5
v3
v5
preimage
v2
v4
v2
v4
G-v2
v1
v3
v5
graph G
v3
v5
G-v1
G-v4
v2
v4
v1
v2
v1
v5
v3
v4
G-v3
G-v5
Graph Reconstruction Problem
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Input: multi-set D whose has n graph with n-1 vertices
Question： Is there a preimage whose deck is D?
Input: D
Unlabeled Graphs
Graph Reconstruction Conjecture
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For any multi-set D of graphs with n-1 vertices,
there is at most 1 preimage whose deck is D (n≧3).
Input: D
Graph G
Different graph of G
Graph Reconstruction Conjecture
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Proposed by Ulam and Kelly [1941]

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Open problem
Reconstructible graph classes (the conjecture is true)

regular graphs, trees, disconnected graphs, etc.
Related research
 Reconstructible

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degree sequence, chromatic number, etc.
Many graph isomorphism-related complexity results

Isomorphism problem is not easier than reconstruction problem.
Naive Reconstruction Algorithm
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Select G∈D and add a vertex v to G.
Add edges incident to v (GN(v)).
Construct the deck DN(v) of GN(v).
Check that DN(v) is equal to D (Deck Checking).

If DN(v) = D then GN(v) is preimage of D.
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Else goto 2.
Input: D
Deck of GN(v)
v
Graph GN(v)
Naive Reconstruction Algorithm
1.
2.
3.
4.
Select G∈D and add a vertex v to G.
Add edges incident to v (GN(v)).
Construct the deck DN(v) of GN(v).
Check that DN(v) is equal to D (Deck Checking).

If DN(v) = D then GN(v) is preimage of D.

Else goto 2.
Input: D
Deck of GN(v)
≠
D is not a deck of GN(v)
v
Graph GN(v)
Naive Reconstruction Algorithm
1.
2.
3.
4.
Select G∈D and add a vertex v to G.
Add edges incident to v (GN(v)).
Construct the deck DN(v) of GN(v).
Check that DN(v) is equal to D (Deck Checking).

If DN(v) = D then GN(v) is preimage of D.

Else goto 2.
Deck of GN(v)
Input: D
=
D is a deck of GN(v)
v
Graph GN(v)
Naive Reconstruction Algorithm
Select G∈D and add a vertex v to G.
Add edges incident to v (GN(v)).
Construct the deck DN(v) of GN(v).
Check that DN(v) is equal to D (Deck Checking).

If DN(v) = D then GN(v) is preimage of D.

Else goto 2.
1.
2.
3.
4.
Exponential
Polynomial
time
Isomorphism
This algorithm is very slow!
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Polynomial time algorithms
 Input:

restrict the graphs in multi-set D
The input graphs can solve the isomorphism problem in
polynomial time.
GI-complete: the isomorphism problem is as hard as on general graphs
Our Contribution
GI-complete
Perfect graphs
HHD-free graphs
Comparability graphs
Chordal graphs
GI can be solved in
polynomial time
Kiyomi, et al. 2009
Interval graphs
Distancehereditary graphs
Permutation graphs
Exist reconstruction algorithms
The conjecture is true. Proper interval graphs
Tree
Threshold graphs
Reconstruction Problem on Permutation Graphs
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Input: multi-set D
Question: Is there a permutation graph whose deck is D?
Input: D
・
・
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Permutation graph
Permutation Graphs
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A graph is called a permutation graph
if the graph has a line representation.
(also, permutation diagraph)
1
2
3
4
5
6
1
6
4
3
3
6
4
1 5
2
Line representation
2
5
Permutation graph
Permutation Graphs
Lemma 1
Induced subgraphs of a permutation graph are
permutation graphs.
1
2
3
4
5
6
1
6
4
3
6
4
1 5
2
Line Representation
3
2
5
Permutation graph
A preimage G is a permutation graph
⇒ each graph of the deck of G is a permutation graph.
Reconstruction Problem on Permutation Graphs
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Input: multi-set D
 Each graph G∈D is a permutation graph
Question: Is there a permutation graph whose deck is D?
Permutation graphs
Input: D
・
・
・
Reconstruction Algorithm for
Permutation Graphs?
 Adding
a line segment to line representation of Gi in Deck.
There exist exponentially
many line representations.
Input: D
Graph Gi
O(n2) time?
・
・
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Unique line representation
Unique Line Representation
Lemma 2 [T. Ma and J. Spinrad, 1994]
A permutation graph G that is a prime with respect to
modular decomposition has a unique representation.
Input: D
Graph G
・
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O(n2) time
Modular Decomposition
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A module M is a set of vertices
s.t. vertices of V-M are adjacent to either all vertices of M,
or no vertex of M.
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Module M is trivial if M=φ, M=V, or |M|=1.
G is a prime if G contains only trivial modules.
Not prime
Prime
Unique Line Representation
Lemma 2 [T. Ma and J. Spinrad, 1994]
A permutation graph G that is a prime with respect to
modular decomposition has a unique representation.
Lemma 3 [J.H. Schmerl, W.T. Trotter, 1993]
Let a graph G is a prime.
There is a vertex v s.t. G-v is a prime
⇔ G is not isomorphic to H2n or, H2n
x1
x2
Prime
Prime
y1
y2
・
・
xi ・
・
・
x ・
n
・
・
・ y
i
・
・
・y
n
Graph H2n
Algorithm (a preimage is a prime)
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Foreach graph G in D do
If G is a prime then
Add a line segment to the line representation of G.
If there is no prime then
Check that a preimage is H2n, or H2n

When a preimage is not a prime

By using modular decomposition tree,
we reduce the problem to “the prime case”.
Overlap: M1  M 2   , M1 \ M 2   , and M 2 \ M1  
Modular Decomposition Tree
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A module M is set of vertices

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s.t. vertices of V-M are adjacent to either all vertices of M, or no vertex of M.
A module M is strong if M does not overlap any other modules.
Modular decomposition tree (strong modules M1, M2)
 M1 is an ancestor of M2 ⇔ M1 contains M2
M4
M1
M1
M5
M2
M3
M2
M3
M4
M5
Modular Decomposition and Line Representation
M4
M1
M2
M1
M2
M1 M3
M3
M4
M2
M5M5
M3
M53
M
M4 M1 MM
2 5 M3
M4
M5
A line representation of a minimal strong module is unique.
Algorithm (Preimage is not a prime)
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Foreach graph G∈D do
Construct modular decomposition of G.
Foreach minimal strong module M do
Add a line segment to the line representation of M.
Check that a preimage has H2n, H2n, or twins.
Conclusions and Future Works
Perfect graph
GI-complete
HHD-free graph
Comparability graph
Chordal graph
GI can be solved in polynomial time
Circle graph
Circular-arc graph
DistanceIs the
Interval graph
Hereditary graph
conjecture true?
Permutation graph
Propose polynomial time algorithms
Exist reconstruction algorithms
The conjecture is true. Proper Interval graph
Tree
Threshold graph