Bipartite Permutation Graphs are Reconstructible

1. Bipartite Permutation Graphs are Reconstructible ToshikiSaitoh (ERATO) Joint work with Masashi Kiyomi (JAIST) and RyuheiUehara (JAIST) COCOA 2010 18-20/Dec/2010

2. v1 v2 v2 v4 v3 v5 v1 v3 v5 v2 v4 v1 v4 v2 v4 v1 v2 Graph G v3 v3 v5 v3 v5 v1 v5 v4 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 Deck of G Preimage G-v2 G-v4 G-v1 G-v5 G-v3

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

4. 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.

5. Bipartite Permutation Graphs Permutation graph: graph that has a permutation diagram. Bipartite permutation graph: permutation graph that is bipartite. 1 2 3 4 5 6 Permutation graph 1 6 3 6 4 1 5 2 Permutation diagram 1 2 4 7 4 1 2 3 4 5 6 7 8 3 2 5 3 5 6 8 3 5 6 1 2 8 4 7 Bipartite permutation graph Permutation diagram

6. Bipartite Permutation Graphs Lemma 1 Induced subgraphs of a bipartite permutation graph are bipartite permutation graphs. 1 2 3 4 5 6 7 8 1 2 4 7 3 5 6 8 3 5 6 1 2 8 4 7 A preimageG is a bipartite permutation graph Each graph in the deck of G is a bipartite permutation graph.

7. 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 Rotation Vertical-flip Vertical-flip horizontal-flip Each permutation diagramof a graph in the deck can be obtained by removing one segment.

8. Theorem Bipartite permutation graphs are reconstructible. There are O(n2) candidates. We show only one candidate is valid. • 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.

9. Choosing Valid Candidate deg(w) = 2 v deg(v): p-1 → p Using the deg(w) we have only one choice. • Using the degree of a polar vertex of the preimage. • Polar vertex: Left-most or right-most segment • Let a vertex vbe a polar vertex of the preimageG and deg(v) = pin G • There is a graph in the deck obtained byremoving a vertex wadjacent 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]

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

11. 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 n2.