Subgraph Isomorphism in Graph Classes
Toshiki Saitoh
ERATO, Minato Project, JST
Joint work with
Yota Otachi, Shuji Kijima, and Takeaki Uno
The 14th Korea-Japan Joint Workshop on Algorithms and Computation
8-9, July, 2011 (Busan, Korea)
Subgraph Isomorphism Problem
 Input: Two graphs G=(VG, EG) and H=(VH, EH)
 |VH|≦|VG| and |EH|≦|EG|
 Question: Is H a subgraph isomorphic to G?
 Is there an injective map f from VH to VG
 {f(u), f(v)}∈EG holds for any {u, v}∈EH
Example
Graph G
Yes
Graph H1
No
Graph H2
Subgraph Isomorphism Problem
 Input: Two graphs G=(VG, EG) and H=(VH, EH)
 |VH|≦|VG| and |EH|≦|EG|
 Question: Is H a subgraph isomorphic to G?
 Is there an injective map f from VH to VG
 {f(u), f(v)}∈EG holds for any {u, v}∈EH
Application
•LSI design
•Pattern recognition
•Bioinfomatics
•Computer vision, etc.
Subgraph Isomorphism Problem
 NP-complete in general


Contains maximum clique, Hamiltonian path, etc.
Graph classes
o Outerplanar graphs
o Cographs
 Polynomial time
 k-connected partial k-tree
 Tree (1-connected partial 1-tree)
 H is forest and G is tree ⇒ NP-hard
 2-connected series-parallel graphs
Our results
G, H: Connected
G, H∈Graphclass C
Perfect
HHD-free
Comparability
Chordal
Bipartite
NP-hard
(Known)
Distance-hereditary
Cograph
Permutation
Interval
Bipartite permutation
Ptolemaic
Proper interval
Trivially perfect
Co-chain
Threshold
NP-hard
Polynomial
Chain
Polynomial
(Known)
Tree
Proper Interval Graphs (PIGs)
 Have proper interval representations
 Each interval corresponds to a vertex
 Two intervals intersect ⇔
corresponding two vertices
are adjacent
 No interval properly contains another
Proper interval graph and its proper interval representation
Characterization of PIGs
 Every PIG has at most 2 Dyck paths.
 Two PIGs G and H are isomorphic
⇔ the Dyck path of G is equal to the Dyck path of H.
 A maximum clique of a PIG G corresponds to
a highest point of a
Dyck path.
 If a PIG G is connected, G contains a Hamilton path.
We thought that
the subgraph isomorphism problem of PIGs is easy.
But,
NP-complete!
Problem
Connected
 Input: Two proper interval graphs G=(VG, EG) and H=(VH, EH)
= |VG| and |EH| < |EG|
|V
|VHH| |≦|V
 Question: Is H a subgraph isomorphic to G?
NP-complete
Reduction from 3-partition problem
3-Partition
 Input: Set A of 3m elements, a bound B∈Z+, and a size aj∈Z+ for each j∈A
 Each aj satisfies that B/4 < aj < B/2
 Σj∈A aj = mB
 Question: Can A be partitioned into m disjoint sets A(1), ... , A(m),
for 1≦i≦m, Σj∈A(i) aj = B
Proof (G and H are disconnected)
Cliques of size B
G
…
m
…
…
…
…
…
Proof (G and H are disconnected)
Cliques of size B
G
…
m
H
Cliques
…
a1
a2
a3
a3m
m>2
Proof (G and H are disconnected)
Cliques of size BM+6m2
BM+3m2
BM+3m2
G
…
…
…
…
…
…
…
3m2
(M=7m3)
H
…
a 1M
a2M
a3M
a3mM
m>2
Proof (G is connected)
Cliques of size BM+6m2
G
…
…
…
…
…
…
…
…
(M=7m3)
…
3m2
Cliques of size 6m2
H
…
a 1M
a2M
a3M
a3mM
m>2
Proof (G is connected)
Cliques of size BM+6m2
G
…
…
…
…
…
…
…
…
…
…
…
…
…
3m2
…
Cliques of size 6m2
(M=7m3)
BM
…
…
…
3m2
…
…
…
…
m>2
Proof (G is connected)
Cliques of size BM+6m2
G
…
…
…
…
…
…
…
…
(M=7m3)
…
3m2
Cliques of size 6m2
H
…
a 1M
a2M
a3M
a3mM
m>2
Proof (G and H are connected)
Cliques of size BM+6m2
G
H
Paths of length m
…
a 1M
…
a2M
…
Cliques of size 6m2
…
(M=7m3)
…
…
…
…
…
…
…
3m2
…
a3M
a3mM
m>2
Proof (G and H are connected)
…
paths
…
…
…
…
a1M
…
…
(M=7m3)
H
Paths of length m
…
a 1M
…
a2M
…
a3M
a3mM
m>2
Proof (G and H are connected)
Cliques of size BM+6m2
G
H
Paths of length m
…
a 1M
…
a2M
…
Cliques of size 6m2
…
(M=7m3)
…
…
…
…
…
…
…
3m2
…
a3M
a3mM
m>2
Proof (|VG|=|VH|)
Cliques of size BM+6m2
G
H
Paths of length m
…
Cliques of size 6m2
…
(M=7m3)
…
…
…
…
…
…
…
3m2
6m3-m2-3m+2
…
…
a 1M
…
a2M
…
a3M
a3mM
Conclusion
G, H: Connected
G, H∈Graphclass C
Perfect
HHD-free
Comparability
Chordal
NP-hard
(Known)
Bipartite
Distance-hereditary
Cograph
Permutation
Interval
Bipartite permutation
Proper interval
Ptolemaic
Trivially perfect
NP-hard
Polynomial
Co-chain
Chain
Polynomial
(Known)
Tree
Threshold