Reconstruction Algorithm for Permutation Graphs Masashi Kiyomi, Toshiki Saitoh, and Ryuhei Uehara School of Information Science Japan Advanced Institute of Science and Technology

Graph Reconstruction Problem Deck of Graph G=(V, E): multi-set {G-v | v ∈ V} Preimage of multi-set D: a graph whose deck is D v1v1 v2v2 v3v3 v5v5 v4v4 graph G v2v2 v3v3 v5v5 v4v4 G-v 1 v1v1 v3v3 v5v5 v4v4 G-v 2 v1v1 v2v2 v5v5 v4v4 G-v 3 G-v 4 v1v1 v2v2 v3v3 v4v4 G-v 5 deck of G preimage v1v1 v2v2 v3v3 v5v5

Graph Reconstruction Problem Input: multi-set D whose has n graph with n-1 vertices Question ： Is there a preimage whose deck is D? Input: D Unlabeled Graphs

Graph Reconstruction Conjecture For any multi-set D of graphs with n-1 vertices, there is at most 1 preimage whose deck is D (n ≧ 3). Different graph of G Input: D Graph G

Graph Reconstruction Conjecture Proposed by Ulam and Kelly [1941]  Open problem Reconstructible graph classes (the conjecture is true)  regular graphs, trees, disconnected graphs, etc. Related research Reconstructible  degree sequence, chromatic number, etc. Many graph isomorphism-related complexity results  Isomorphism problem is not easier than reconstruction problem.

Naive Reconstruction Algorithm Input: D v Graph G N(v) Deck of G N(v) 1. Select G ∈ D and add a vertex v to G. 2. Add edges incident to v (G N(v) ). 3. Construct the deck D N(v) of G N(v). 4. Check that D N(v) is equal to D (Deck Checking).  If D N(v) = D then G N(v) is preimage of D.  Else goto 2.

Naive Reconstruction Algorithm Input: D v Graph G N(v) ≠ D is not a deck of G N(v) 1. Select G ∈ D and add a vertex v to G. 2. Add edges incident to v (G N(v) ). 3. Construct the deck D N(v) of G N(v). 4. Check that D N(v) is equal to D (Deck Checking).  If D N(v) = D then G N(v) is preimage of D.  Else goto 2. Deck of G N(v)

Naive Reconstruction Algorithm 1. Select G ∈ D and add a vertex v to G. 2. Add edges incident to v (G N(v) ). 3. Construct the deck D N(v) of G N(v). 4. Check that D N(v) is equal to D (Deck Checking).  If D N(v) = D then G N(v) is preimage of D.  Else goto 2. Input: D v Graph G N(v) = D is a deck of G N(v) Deck of G N(v)

Naive Reconstruction Algorithm Isomorphism Exponential This algorithm is very slow! Polynomial time algorithms  Input: restrict the graphs in multi-set D The input graphs can solve the isomorphism problem in polynomial time. Polynomial time 1. Select G ∈ D and add a vertex v to G. 2. Add edges incident to v (G N(v) ). 3. Construct the deck D N(v) of G N(v). 4. Check that D N(v) is equal to D (Deck Checking).  If D N(v) = D then G N(v) is preimage of D.  Else goto 2.

Our Contribution Permutation graphs Tree Distance- hereditary graphs Chordal graphs Interval graphs HHD-free graphs Perfect graphs Proper interval graphs Threshold graphs GI-complete: the isomorphism problem is as hard as on general graphs Comparability graphs GI-complete GI can be solved in polynomial time Exist reconstruction algorithms The conjecture is true. Kiyomi, et al. 2009

Reconstruction Problem on Permutation Graphs Input: multi-set D Question: Is there a permutation graph whose deck is D? Input: D Permutation graph

Permutation Graphs A graph is called a permutation graph if the graph has a line representation Line representation Permutation graph (also, permutation diagraph)

Permutation Graphs Line Representation Permutation graph A preimage G is a permutation graph ⇒ each graph of the deck of G is a permutation graph. Lemma 1 Induced subgraphs of a permutation graph are permutation graphs.

Reconstruction Problem on Permutation Graphs Input: multi-set D  Each graph G ∈ D is a permutation graph Question: Is there a permutation graph whose deck is D? Input: D Permutation graphs

Reconstruction Algorithm for Permutation Graphs? Adding a line segment to line representation of G i in Deck. Input: D Graph G i O(n 2 ) time? Unique line representation There exist exponentially many line representations.

Unique Line Representation Input: D Graph G O(n 2 ) time Lemma 2 [T. Ma and J. Spinrad, 1994] A permutation graph G that is a prime with respect to modular decomposition has a unique representation.

Modular Decomposition A module M is a set of vertices s.t. vertices of V-M are adjacent to either all vertices of M, or no vertex of M. Module M is trivial if M=φ, M=V, or |M|=1. G is a prime if G contains only trivial modules. Prime Not prime

Unique Line Representation Graph H 2n x1x1 x2x2 xixi xnxn y1y1 y2y2 yiyi ynyn Prime Lemma 2 [T. Ma and J. Spinrad, 1994] A permutation graph G that is a prime with respect to modular decomposition has a unique representation. Lemma 3 [J.H. Schmerl, W.T. Trotter, 1993] Let a graph G is a prime. There is a vertex v s.t. G-v is a prime ⇔ G is not isomorphic to H 2n or, H 2n

Algorithm (a preimage is a prime) 1. Foreach graph G in D do 2. If G is a prime then 3. Add a line segment to the line representation of G. 4. If there is no prime then 5. Check that a preimage is H 2n, or H 2n When a preimage is not a prime  By using modular decomposition tree, we reduce the problem to “the prime case”.

Modular Decomposition Tree A module M is set of vertices  s.t. vertices of V-M are adjacent to either all vertices of M, or no vertex of M. A module M is strong if M does not overlap any other modules. Modular decomposition tree (strong modules M 1, M 2 )  M 1 is an ancestor of M 2 ⇔ M 1 contains M 2 M1M1 M2M2 M3M3 M4M4 M5M5 M1M1 M2M2 M3M3 M4M4 M5M5

M4M4 M5M5 M 1 M 2 M 3 Modular Decomposition and Line Representation A line representation of a minimal strong module is unique. M1M1 M2M2 M3M3 M1M1 M2M2 M3M3 M5M5 M4M4 M 4 M 5 M3M3 M5M5 M1M1 M2M2 M3M3 M4M4 M5M5

Algorithm (Preimage is not a prime) 1. Foreach graph G ∈ D do 2. Construct modular decomposition of G. 3. Foreach minimal strong module M do 4. Add a line segment to the line representation of M. 5. Check that a preimage has H 2n, H 2n, or twins.

Conclusions and Future Works Permutation graph Tree Distance- Hereditary graph Chordal graph Interval graph HHD-free graph Perfect graph Proper Interval graph Threshold graph Comparability graph Circle graph Circular-arc graph Is the conjecture true? GI-complete GI can be solved in polynomial time Exist reconstruction algorithms The conjecture is true. Propose polynomial time algorithms