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Exploring Subgraph Isomorphism in Graph Classes: NP-Completeness and Applications

This paper examines the Subgraph Isomorphism Problem, focusing on its complexities in various graph classes. We analyze input graphs G and H, determining whether H is a subgraph isomorphic to G. The problem remains NP-complete in general, yet we highlight specific cases such as proper interval graphs and their characteristics, including proper interval representations. Applications in LSI design, bioinformatics, and pattern recognition are discussed, alongside the fundamental NP-hardness of various graph configurations.

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Exploring Subgraph Isomorphism in Graph Classes: NP-Completeness and Applications

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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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