Subgraph Isomorphism Problem for Graph Classes

Toshiki SaitohERATO, Minato Discrete Structure Manipulation System Project, JST Graph Classes and Subgraph IsomorphismJoint work with Yota Otachi, Shuji Kijima, and Takeaki Uno20101119Subgraph Isomorphism ProblemInput: 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}EHExampleGraph GGraph H1Graph H2YesNoSubgraph Isomorphism ProblemInput: 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}EHApplicationLSI designPattern recognitionBioinfomaticsComputer vision, etc.3Known ResultNP-complete in generalContains maximum clique, Hamiltonian path, Isomorphism problem etc.Graph classesOuterplanar graphsCographs

Polynomial time algorithmsk-connected partial k-treesTreeH is forest NP-hard2-connected series-parallel graphsGraph ClassesChordalIntervalDistance-hereditaryPtolemaicCographComparabilityPermutationPerfectBipartiteHHD-freeTrivially perfectProper intervalThresholdBipartite permutationChainCo-chainNP-hardNP-hardTreeG, H: ConnectedG, HGraphclass C5Proper 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 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 of 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 BGmHa1a2a3a3mCliquesProof (G and H are disconnected)GHma1MCliques of size BM(M=m10)a2Ma3Ma3mMProof (G and H are disconnected)GHa1MCliques of size BM+6m2(M=m10)a2Ma3Ma3mMm>23m2Proof (G is connected)GHa1MCliques of size BM+6m2(M=m10)a2Ma3Ma3mMm>23m2Cliques of size 6m2Proof (G is connected)GCliques of size BM+6m2(M=m10)m>23m2Cliques of size 6m2Proof (G is connected)GHa1MCliques of size BM+6m2(M=m10)a2Ma3Ma3mMm>23m2Cliques of size 6m2Proof (G and H are connected)GHa1MCliques of size BM+6m2(M=m10)a2Ma3Ma3mMm>23m2Cliques of size 6m2Paths of length mProof (G and H are connected)Ha1M(M=m10)a2Ma3Ma3mMm>2Paths of length mProof (G and H are connected)GHa1MCliques of size BM+6m2(M=m10)a2Ma3Ma3mMm>23m2Cliques of size 6m2Paths of length mProof (|VG|=|VH|)GHa1MCliques of size BM+6m2(M=m10)a2Ma3Ma3mMm>23m2Cliques of size 6m2Paths of length m6m3-m2-3m+2Graph ClassesChordalIntervalDistance-hereditaryPtolemaicCographComparabilityPermutationPerfectBipartiteHHD-freeTrivially perfectProper intervalThresholdBipartite permutationChainCo-chainNP-hardNP-hardTreeG, H: connectedG, HGraphclass C20Threshold GraphsA graph G=(V, E) is a thresholdThere are a real number S and a real vertex weight w(v) such that (u,v) E w(u)+w(v)SG=(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)Graph Gv1v2v3v4v5v6v7Degree sequence: 6 6 4 3 3 2 2v1, v2, v3, v4, v5, v6, v7N[v1]N[v2]N[v3]N(v4)N(v5)N(v6)N(v7)Lemma [Hammer, et al. 78]N(v): neighbor set of vN[v]:closed neighbor set of vPolynomial Time Algorithm Finds degree sequences of G and HG : ( d(v1), d(v2), , d(vn) )H : ( d(u1), d(u2), , d(un) ) for i=1 to n do if d(vi) < d(ui) then return No! return Yes!Graph GGraph H1Graph H2YesNoG : 6 6 4 3 3 2 2H1: 6 5 3 3 2 2 1G : 6 6 4 3 3 2 2H2: 5 5 5 3 3 3Our ResultsChordalIntervalDistance-hereditaryPtolemaicCographComparabilityPermutationPerfectBipartiteHHD-freeTrivially perfectProper intervalThresholdBipartite permutationChainCo-chainNP-hardNP-hardTreeG, H: connectedG, HGraphclass C23