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Graph Classes and Subgraph Isomorphism

Graph Classes and Subgraph Isomorphism. Toshiki Saitoh ERATO, Minato Discrete Structure Manipulation System Project, JST . Joint work with Yota Otachi , Shuji Kijima, and Takeaki Uno. アルゴリズム研究会 2010 年 1 1 月 19 日. Subgraph Isomorphism Problem.

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Graph Classes and Subgraph Isomorphism

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  1. Graph Classes andSubgraph Isomorphism ToshikiSaitoh ERATO, Minato Discrete Structure Manipulation System Project, JST Joint work with YotaOtachi, Shuji Kijima, and Takeaki Uno アルゴリズム研究会2010年11月19日

  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 of G? • 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 of 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. Known Result • NP-complete in general • Containsmaximum clique, Hamiltonian path, Isomorphism problem etc. • Graph classes • Outerplanar graphs • Cographs • Polynomial time algorithms • k-connected partial k-trees • Tree • H is forest ⇒ NP-hard • 2-connected series-parallel graphs

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

  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 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 of 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. Proof (G and H are disconnected) Cliques of size BM G … … m (M=m10) H … … a3M a3mM a1M a2M

  12. m>2 Proof (G and H are disconnected) Cliques of size BM+6m2 G … … … … … … … 3m2 (M=m10) H … a3M a3mM a1M a2M

  13. m>2 Proof (Gis connected) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=m10) H … a3M a3mM a1M a2M

  14. m>2 Proof (Gis connected) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=m10) … … … … … … … … … … … …

  15. m>2 Proof (Gis connected) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=m10) H … a3M a3mM a1M a2M

  16. m>2 Proof (G and H are connected) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=m10) H Paths of length m … … … a3M a3mM a1M a2M

  17. m>2 Proof (G and H are connected) … … … (M=m10) H Paths of length m … … … … … … … a3M a3mM a1M a2M

  18. m>2 Proof (G and H are connected) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=m10) H Paths of length m … … … a3M a3mM a1M a2M

  19. m>2 Proof (|VG|=|VH|) Cliques of size BM+6m2 G … … … … … … … … … 3m2 Cliques of size 6m2 (M=m10) H Paths of length m 6m3-m2-3m+2 … … … … a3M a3mM a1M a2M

  20. G, H: connected G, H∈GraphclassC Perfect Graph Classes HHD-free Comparability Chordal Distance-hereditary Bipartite Cograph NP-hard Ptolemaic Permutation Interval Bipartite permutation Proper interval Trivially perfect NP-hard Chain Tree Co-chain Threshold

  21. N(v): neighbor set of v N[v]:closed neighbor set of v Threshold Graphs • A graph G=(V, E) is a threshold • There are a real number S and a real vertex weight w(v) such that (u,v) ∈E⇔w(u)+w(v)≧S Lemma [Hammer, et al. 78] • G=(V, E): graph, (d(v1), d(v2), …, d(vn)): degree sequence of G. • G is a threshold • ⇔ N[v1]⊇N[v2]⊇… ⊇N[vi]⊇N(vi+1)⊇… ⊇ N(vn) v1 v7 v2 Degree sequence: 6 6 4 3 3 2 2 v1, v2, v3, v4, v5, v6, v7 v6 v3 N[v1]⊇N[v2]⊇N[v3]⊇N(v4)⊇N(v5)⊇N(v6)⊇N(v7) v5 v4 Graph G

  22. Polynomial Time Algorithm • Finds degree sequences of G and H • G : ( d(v1), d(v2), …, d(vn) ) • H : ( d(u1), d(u2), …, d(un’) ) • fori=1ton’do • if d(vi) < d(ui) then return No! • return Yes! Yes No Graph H2 Graph H1 G: 6 6 4 3 3 2 2 H2: 5 5 5 3 3 3 G: 6 6 4 3 3 2 2 H1: 6 5 3 3 2 2 1 Graph G

  23. G, H: connected G, H∈GraphclassC Perfect Our Results HHD-free Comparability Chordal Distance-hereditary Bipartite Cograph NP-hard Ptolemaic Permutation Interval Bipartite permutation Proper interval Trivially perfect NP-hard Chain Tree Threshold Co-chain

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