1. Toshiki Saitoh ERATO, Minato Project, JST Subgraph Isomorphism in Graph Classes Joint work with Yota Otachi, Shuji Kijima, and Takeaki Uno The 14 th Korea-Japan Joint Workshop on Algorithms and Computation 8-9, July, 2011 (Busan, Korea)

2. Subgraph Isomorphism Problem Input: Two graphs G=(V G, E G ) and H=(V H, E H ) |V H | ≦ |V G | and |E H | ≦ |E G | Question: Is H a subgraph isomorphic to G? Is there an injective map f from V H to V G {f(u), f(v)} ∈ E G holds for any {u, v} ∈ E H Example Graph G Graph H 1 Graph H 2 YesNo

3. Subgraph Isomorphism Problem Input: Two graphs G=(V G, E G ) and H=(V H, E H ) |V H | ≦ |V G | and |E H | ≦ |E G | Question: Is H a subgraph isomorphic to G? Is there an injective map f from V H to V G {f(u), f(v)} ∈ E G holds for any {u, v} ∈ E H Application LSI design Pattern recognition Bioinfomatics Computer vision, etc.

4. NP-complete in general Contains maximum clique, Hamiltonian path, etc. Graph classes oOuterplanar graphs oCographs 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 Subgraph Isomorphism Problem

5. Our results Chordal Interval Distance-hereditary Ptolemaic Cograph Comparability Permutation Perfect Bipartite HHD-free Trivially perfectProper interval Threshold Bipartite permutation Chain Co-chain NP-hard (Known) NP-hard Tree G, H: Connected G, H ∈ Graphclass C G, H: Connected G, H ∈ Graphclass C Polynomial (Known) Polynomial

6. 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

7. 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. NP-complete! But,

8. Problem Input: Two proper interval graphs G=(V G, E G ) and H=(V H, E H ) |V H | ≦ |V G | and |E H | < |E G | Question: Is H a subgraph isomorphic to G? |V H | = |V G | Connected NP-complete Reduction from 3-partition problem 3-Partition Input: Set A of 3m elements, a bound B ∈ Z +, and a size a j ∈ Z + for each j ∈ A Each a j satisfies that B/4 < a j < B/2 Σ j ∈ A a j = mB Question: Can A be partitioned into m disjoint sets A(1),..., A(m), for 1 ≦ i ≦ m, Σ j ∈ A(i) a j = B

9. Proof ( G and H are disconnected) ………… … … Cliques of size B G m

10. Proof ( G and H are disconnected) … Cliques of size B G m H … a1a1 a2a2 a3a3 a3ma3m Cliques

11. Proof ( G and H are disconnected) … G H … a1Ma1M Cliques of size BM+6m 2 (M=7m 3 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 ……… … … 3m23m2 … BM+3m 2

12. Proof ( G is connected) G H … a1Ma1M Cliques of size BM+6m 2 (M=7m 3 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 … ……… … …… 3m23m2 …… Cliques of size 6m 2

13. Proof ( G is connected) … G Cliques of size BM+6m 2 (M=7m 3 ) m>2 ……… … …… 3m23m2 …… Cliques of size 6m 2 ……… … … … … … … … … … 3m23m2 BM

14. Proof ( G is connected) G H … a1Ma1M Cliques of size BM+6m 2 (M=7m 3 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 … ……… … …… 3m23m2 …… Cliques of size 6m 2

15. Proof ( G and H are connected) G H … a1Ma1M Cliques of size BM+6m 2 (M=7m 3 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 Cliques of size 6m 2 …… Paths of length m 3m23m2 … …… … ………… …

16. Proof ( G and H are connected) H … a1Ma1M (M=7m 3 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 …… Paths of length m ……… … … …… a1Ma1M paths

17. Proof ( G and H are connected) G H … a1Ma1M Cliques of size BM+6m 2 (M=7m 3 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 3m23m2 Cliques of size 6m 2 …… Paths of length m … …… … ………… …

18. Proof ( |V G |=|V H | ) G H … a1Ma1M Cliques of size BM+6m 2 (M=7m 3 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 3m23m2 Cliques of size 6m 2 …… Paths of length m … 6m 3 -m 2 -3m+2 … …… … ………… …

19. Conclusion Chordal Interval Distance-hereditary Ptolemaic Cograph Comparability Permutation Perfect Bipartite HHD-free Trivially perfectProper interval Threshold Bipartite permutation Chain Co-chain NP-hard (Known) NP-hard Tree G, H: Connected G, H ∈ Graphclass C G, H: Connected G, H ∈ Graphclass C Polynomial (Known) Polynomial