Subgraph Isomorphism Problem for Graph Classes

Toshiki SaitohERATO, Minato Project, JST Subgraph Isomorphism in Graph ClassesJoint work with Yota Otachi, Shuji Kijima, and Takeaki UnoThe 14th Korea-Japan Joint Workshop on Algorithms and Computation8-9, July, 2011 (Busan, Korea)1Subgraph Isomorphism ProblemInput: 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}EHExampleGraph GGraph H1Graph H2YesNo2Subgraph Isomorphism ProblemInput: 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}EHApplicationLSI designPattern recognitionBioinfomaticsComputer vision, etc.3NP-complete in generalContains maximum clique, Hamiltonian path, etc.Graph classesOuterplanar graphsCographs

Polynomial timek-connected partial k-treeTree (1-connected partial 1-tree)H is forest and G is tree NP-hard2-connected series-parallel graphsSubgraph Isomorphism Problem4Our resultsChordalIntervalDistance-hereditaryPtolemaicCographComparabilityPermutationPerfectBipartiteHHD-freeTrivially perfectProper intervalThresholdBipartite permutationChainCo-chainNP-hard (Known)NP-hardTreeG, H: ConnectedG, HGraphclass CPolynomial(Known)Polynomial5Proper Interval Graphs (PIGs)Have proper interval representationsEach interval corresponds to a vertexTwo intervals intersect corresponding two vertices are adjacentNo interval properly contains anotherProper interval graph and its proper interval representationCharacterization of PIGsEvery 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, ProblemInput: 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| ConnectedNP-completeReduction from 3-partition problem3-PartitionInput: Set A of 3m elements, a bound BZ+, and a size ajZ+ for each jAEach aj satisfies that B/4 < aj < B/2jA aj = mBQuestion: Can A be partitioned into m disjoint sets A(1), ... , A(m), for 1im, jA(i) aj = BProof (G and H are disconnected)Cliques of size BGmProof (G and H are disconnected)Cliques of size BGmHa1a2a3a3mCliques10Proof (G and H are disconnected)GHa1MCliques of size BM+6m2(M=7m3)a2Ma3Ma3mMm>23m2BM+3m2BM+3m211Proof (G is connected)GHa1MCliques of size BM+6m2(M=7m3)a2Ma3Ma3mMm>23m2Cliques of size 6m2Proof (G is connected)GCliques of size BM+6m2(M=7m3)m>23m2Cliques of size 6m23m2BMProof (G is connected)GHa1MCliques of size BM+6m2(M=7m3)a2Ma3Ma3mMm>23m2Cliques of size 6m2Proof (G and H are connected)GHa1MCliques of size BM+6m2(M=7m3)a2Ma3Ma3mMm>2Cliques of size 6m2Paths of length m3m2Proof (G and H are connected)Ha1M(M=7m3)a2Ma3Ma3mMm>2Paths of length ma1MpathsProof (G and H are connected)GHa1MCliques of size BM+6m2(M=7m3)a2Ma3Ma3mMm>23m2Cliques of size 6m2Paths of length mProof (|VG|=|VH|)GHa1MCliques of size BM+6m2(M=7m3)a2Ma3Ma3mMm>23m2Cliques of size 6m2Paths of length m6m3-m2-3m+2ConclusionChordalIntervalDistance-hereditaryPtolemaicCographComparabilityPermutationPerfectBipartiteHHD-freeTrivially perfectProper intervalThresholdBipartite permutationChainCo-chainNP-hard (Known)NP-hardTreeG, H: ConnectedG, HGraphclass CPolynomial(Known)Polynomial19