Bipartite Permutation Graphs are Reconstructible Toshiki Saitoh (ERATO) Joint work with Masashi Kiyomi (JAIST) and Ryuhei Uehara (JAIST) COCOA /Dec/2010

Graph Reconstruction Conjecture 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 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 Multi-set: D Graph G Unlabeled graphs

Graph Reconstruction Conjecture Proposed by Ulam and Kelly [1941]  Open problem Reconstructible graph classes  Reconstructible: Its deck has only one preimage.  regular graphs, trees, disconnected graphs, etc. Our Result Bipartite Permutation Graphs are Reconstructible.

Bipartite Permutation Graphs Permutation graph: graph that has a permutation diagram. Bipartite permutation graph: permutation graph that is bipartite Permutation diagram Permutation graph Permutation diagram Bipartite permutation graph

Bipartite Permutation Graphs Lemma 1 Induced subgraphs of a bipartite permutation graph are bipartite permutation graphs A preimage G is a bipartite permutation graph Each graph in the deck of G is a bipartite permutation graph.

Bipartite Permutation Graphs Lemma 1 Induced subgraphs of a bipartite permutation graph are bipartite permutation graphs. Lemma 2 [Saitoh et al. 2009] There exists at most four permutation diagrams for any connected bipartite permutation graph. horizontal-flip Vertical-flip horizontal-flip Vertical-flip Rotation Each permutation diagram of a graph in the deck can be obtained by removing one segment.

Only show the connected case.  Every disconnected graphs are reconstructible. Main Idea of Proof  Uniquely reconstruct a preimage. By adding a segment uniquely to a permutation diagram of some graph in the deck. Theorem Bipartite permutation graphs are reconstructible. There are O(n 2 ) candidates. We show only one candidate is valid.

Using the degree of a polar vertex of the preimage. Polar vertex: Left-most or right-most segment  Let a vertex v be a polar vertex of the preimage G and deg(v) = p in G There is a graph in the deck obtained by removing a vertex w adjacent to v. Clearly deg(v) = p-1 in the graph.  We know the degree of the removing vertex w. Degree sequence is reconstructible. [Greenwell and Hemminger 73] v deg(v): p-1 → p Using the deg(w) we have only one choice. deg(w) = 2 Choosing Valid Candidate

Finding the Degree of a Polar Vertex Using lemma 3  Choose connected graphs with removing a vertex in Y. There are three possibilities of X-polar degree patterns. pq-1 … … … p-1q … … … We can determine p and q. pq … … … Lemma 3 G=(X, Y, E): Connected bipartite permutation graph. |X| and |Y| are reconstructible.

Conclusion and Future Works Our result  Bipartite permutation graphs are reconstructible. Future works  Are the other graph classes reconstructible? For example, interval graphs, permutation graphs, etc.  The number of preimages are at most n 2.