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Subgraph Isomorphism in Graph Classes

Subgraph Isomorphism in Graph Classes. Toshiki Saitoh ERATO, Minato Project, JST . 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). Subgraph Isomorphism Problem.

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Subgraph Isomorphism in Graph Classes

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

  2. Subgraph Isomorphism Problem • Input: Two graphs G=(VG, EG) and H=(VH, EH) • |VH|≦|VG| and |EH|≦|EG| • Question: Is H a subgraph isomorphic toG? • Is there an injective map f from VH to VG • {f(u), f(v)}∈EG holds for any {u, v}∈EH Example Yes No Graph G Graph H1 Graph H2

  3. 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.

  4. Subgraph Isomorphism Problem • NP-complete in general • Containsmaximum clique, Hamiltonian path, etc. • Graph classes • Outerplanar graphs • 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

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

  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 pointof 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 Connected • Input: Two proper interval graphs G=(VG, EG) and H=(VH, EH) • |VH|≦|VG| and |EH| < |EG| • Question: Is H a subgraph isomorphic to G? |VH| = |VG| NP-complete Reduction from 3-partition problem • 3-Partition • Input: SetA 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∈Aaj = mB • Question: Can A be partitioned intom disjoint sets A(1), ... , A(m), for 1≦i≦m, Σj∈A(i)aj = 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 Cliques … a1 a2 a3 a3m

  11. m>2 Proof (G and H are disconnected) BM+3m2 Cliques of size BM+6m2 BM+3m2 G … … … … … … … 3m2 (M=7m3) H … a3M a3mM a1M a2M

  12. m>2 Proof (Gis connected) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=7m3) H … a3M a3mM a1M a2M

  13. m>2 Proof (Gis connected) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=7m3) BM … … … … … 3m2 … … … … … … …

  14. m>2 Proof (Gis connected) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=7m3) H … a3M a3mM a1M a2M

  15. m>2 Proof (G and H are connected) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=7m3) H Paths of length m … … … a3M a3mM a1M a2M

  16. m>2 Proof (G and H are connected) … a1M … … paths (M=7m3) H Paths of length m … … … … … … … a3M a3mM a1M a2M

  17. m>2 Proof (G and H are connected) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=7m3) H Paths of length m … … … a3M a3mM a1M a2M

  18. m>2 Proof (|VG|=|VH|) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=7m3) H Paths of length m 6m3-m2-3m+2 … … … … a3M a3mM a1M a2M

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

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