Toshiki Saitoh ERATO, Minato Discrete Structure Manipulation System Project, JST Graph Classes and Subgraph Isomorphism Joint work with Yota Otachi, Shuji Kijima, and Takeaki Uno アルゴリズム研究会 2010 年 11 月 19 日

Subgraph Isomorphism Problem Input: Two graphs G=(V G, E G ) and H=(V H, E H ) |V H | ≦ |V G | and |E H | ≦ |E G | Question: Is H a subgraph isomorphic of G? Is there an injective map f from V H to V G {f(u), f(v)} ∈ E G holds for any {u, v} ∈ E H Example Graph G Graph H 1 Graph H 2 YesNo

Subgraph Isomorphism Problem Input: Two graphs G=(V G, E G ) and H=(V H, E H ) |V H | ≦ |V G | and |E H | ≦ |E G | Question: Is H a subgraph isomorphic of G? Is there an injective map f from V H to V G {f(u), f(v)} ∈ E G holds for any {u, v} ∈ E H Application LSI design Pattern recognition Bioinfomatics Computer vision, etc.

Known Result NP-complete in general Contains maximum clique, Hamiltonian path, Isomorphism problem etc. Graph classes oOuterplanar graphs oCographs Polynomial time algorithms k-connected partial k-trees Tree H is forest ⇒ NP-hard 2-connected series-parallel graphs

Graph Classes Chordal Interval Distance-hereditary Ptolemaic Cograph Comparability Permutation Perfect Bipartite HHD-free Trivially perfect Proper interval Threshold Bipartite permutation ChainCo-chain NP-hard Tree G, H: Connected G, H ∈ Graphclass C

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

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 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,

Problem Input: Two proper interval graphs G=(V G, E G ) and H=(V H, E H ) |V H | ≦ |V G | and |E H | < |E G | Question: Is H a subgraph isomorphic of G? |V H | = |V G | Connected NP-complete Reduction from 3-partition problem 3-Partition Input: Set A of 3m elements, a bound B ∈ Z +, and a size a j ∈ Z+ for each j ∈ A Each a j satisfies that B/4 < a j < B/2 Σ j ∈ A a j = mB Question: Can A be partitioned into m disjoint sets A(1),..., A(m), for 1 ≦ i ≦ m, Σ j ∈ A(i) a j = B

Proof ( G and H are disconnected) … Cliques of size B G m ………… …

Proof ( G and H are disconnected) … Cliques of size B G m H … a1a1 a2a2 a3a3 a3ma3m Cliques

Proof ( G and H are disconnected) … G H m … a1Ma1M Cliques of size BM (M=m 10 ) a2Ma2M a3Ma3Ma3mMa3mM … …

Proof ( G and H are disconnected) … G H … a1Ma1M Cliques of size BM+6m 2 (M=m 10 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 ……… … … 3m23m2 …

Proof ( G is connected) … G H … a1Ma1M Cliques of size BM+6m 2 (M=m 10 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 ……… … …… 3m23m2 …… Cliques of size 6m 2

Proof ( G is connected) … G Cliques of size BM+6m 2 (M=m 10 ) m>2 ……… … …… 3m23m2 …… Cliques of size 6m 2 ……… … … … … … … … … …

Proof ( G is connected) … G H … a1Ma1M Cliques of size BM+6m 2 (M=m 10 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 ……… … …… 3m23m2 …… Cliques of size 6m 2

Proof ( G and H are connected) … G H … a1Ma1M Cliques of size BM+6m 2 (M=m 10 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 ……… … …… 3m23m2 …… Cliques of size 6m 2 …… Paths of length m

Proof ( G and H are connected) H … a1Ma1M (M=m 10 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 …… Paths of length m ……… … … ……

Proof ( G and H are connected) … G H … a1Ma1M Cliques of size BM+6m 2 (M=m 10 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 ……… … …… 3m23m2 …… Cliques of size 6m 2 …… Paths of length m

Proof ( |V G |=|V H | ) … G H … a1Ma1M Cliques of size BM+6m 2 (M=m 10 ) a2Ma2M a3Ma3Ma3mMa3mM m>2 ……… … …… 3m23m2 …… Cliques of size 6m 2 …… Paths of length m … 6m 3 -m 2 -3m+2

Graph Classes Chordal Interval Distance-hereditary Ptolemaic Cograph Comparability Permutation Perfect Bipartite HHD-free Trivially perfect Proper interval Threshold Bipartite permutation ChainCo-chain NP-hard Tree G, H: connected G, H ∈ Graphclass C

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 G=(V, E): graph, (d(v 1 ), d(v 2 ), …, d(v n )): degree sequence of G. G is a threshold ⇔ N[v 1 ] ⊇ N[v 2 ] ⊇ … ⊇ N[v i ] ⊇ N(v i+1 ) ⊇ … ⊇ N(v n ) G=(V, E): graph, (d(v 1 ), d(v 2 ), …, d(v n )): degree sequence of G. G is a threshold ⇔ N[v 1 ] ⊇ N[v 2 ] ⊇ … ⊇ N[v i ] ⊇ N(v i+1 ) ⊇ … ⊇ N(v n ) Graph G v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 Degree sequence: v 1, v 2, v 3, v 4, v 5, v 6, v 7 N[v1]⊇N[v2]⊇N[v3]⊇N(v4)⊇N(v5)⊇N(v6)⊇N(v7)N[v1]⊇N[v2]⊇N[v3]⊇N(v4)⊇N(v5)⊇N(v6)⊇N(v7) Lemma [Hammer, et al. 78] N(v): neighbor set of v N[v]:closed neighbor set of v

Polynomial Time Algorithm 1. Finds degree sequences of G and H G : ( d(v 1 ), d(v 2 ), …, d(v n ) ) H : ( d(u 1 ), d(u 2 ), …, d(u n’ ) ) 2. for i=1 to n’ do 3. if d(v i ) < d(u i ) then return No! 4. return Yes! Graph G Graph H 1 Graph H 2 Yes No G : H 1 : G : H 2 :

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