Bipartite Permutation Graphs are Reconstructible

Bipartite Permutation Graphs are Reconstructible Toshiki Saitoh (ERATO) Joint work with Masashi Kiyomi (JAIST) and Ryuhei Uehara (JAIST) COCOA 2010 18-20/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 Deck of G

v1

v4

v1

v2

v3

v5

v3

v5

Preimage v2

v4

v2

v4

G-v2

v1 v3

v5

Graph G

v3

v5

G-v1

G-v4

v2

v4

v1

v2

v1

v5

v3

v4

G-v3

G-v5

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).

Multi-set: D

Graph G

Different graph of 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. 1

2

3

4

5

6

1

6 4

3 6 4 1 5 2 Permutation diagram 

3 2 5 Permutation graph

Bipartite permutation graph: permutation graph that is bipartite. 1 2 3 4 5 6 7 8

3 5 6 1 2 8 4 7

Permutation diagram

1

2

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7

3

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Bipartite permutation graph

Bipartite Permutation Graphs Lemma 1

Induced subgraphs of a bipartite permutation graph are bipartite permutation graphs. 1 2 3 4 5 6 7 8

3 5 6 1 2 8 4 7

1

2

4

7

3

5

6

8

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 Rotation Vertical-flip

Vertical-flip

horizontal-flip Each permutation diagram of a graph in the deck can be obtained by removing one segment.

Theorem Bipartite permutation graphs are reconstructible. 

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. There are O(n2) candidates.

We show only one candidate is valid.

Choosing Valid Candidate 

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(w) = 2

deg(v): p-1 → p Using the deg(w) we have only one choice.

Finding the Degree of a Polar Vertex Lemma 3

G=(X, Y, E): Connected bipartite permutation graph. |X| and |Y| are reconstructible. 

Using lemma 3  Choose connected 

There are three possibilities of X-polar degree patterns.

p …

q-1

…

graphs with removing a vertex in Y.

…

p-1 …

…

p

q …

We can determine p and q.

…

q

…

…

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.