# Reconstructing graphs with vertices of degree $k$ and $k-1$

The Graph Reconstruction Conjecture claims that any simple graph with 3 or more vertices is reconstructible from its "deck" of vertex-deleted subgraphs. (A nice introduction to this problem is at this Wikipedia page.)

My general question: I would be interested in any recent progress on the conjecture. The sources in the Wikipedia article seem to be quite old. (I also have a copy of a Bondy and Hemminger survey from 1977. A more recent article by Ramachandran (in pdf here) is from 2004 but like many works on this conjecture, quickly detours into other reconstruction questions, edge reconstruction, etc.)

A more specific question: since one can reconstruct the degree sequence of a graph then any regular graph can be reconstructed. But graphs with exactly two degrees, $k$ and $k-1$, seem to be quite hard to reconstruct. I would be especially interested in results related to graphs with exactly two degrees, $k$ and $k-1.$

Even more narrowly, can we reconstruct graphs in which all vertices have degree 2 or 3? (Apparently Kocay worked on that in the early 1980s, says Ramachandran.) Surely "small" graphs of degrees {2,3} are accessible to modern computers so there may now be a place for fertile exploration on this problem?

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Do we know the number of vertices with degree $2$ and $3$? Recently, Adryan Bondy(I think) published a survey about open problems in graph theory. The first problem is about this conjecture. But this is very interesting problem. –  Shahrooz Dec 18 '11 at 9:58
The whole degree sequence can be determined from the deck. –  Brendan McKay Dec 18 '11 at 15:02
I do not think there has been much (any?) recent progress. Asking Bill Kocay would be a good idea. –  Chris Godsil Dec 18 '11 at 15:50

I have a paper on the Reconstruction Conjecture. It was published in an Elsevier journal dedicated to Discrete Mathematics. Its available online since 2007:

Kia Dalili, Sara Faridi and Will Traves. Note: The Reconstruction Conjecture and edge ideals. Discrete Mathematics 308(10), pp. 2002–2010, 2008. (MathSciNet review)

Just type Kia Dalili in the Author field for Search and it will show up.

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Thanks. (The reconstruction conjecture and edge ideals, Dalili, et. al. Discrete Mathematics (Elsevier) v. 308, no. 10 (2008-01-01) p. 2002-2010. ISSN: 0012-365X.) I found a pdf by searching on the title. –  Ken W. Smith Dec 18 '11 at 14:01

Not many people work on the classical reconstruction conjecture these days, probably because only very difficult subproblems remain. The only recent good result I am aware of is this one by Brignall, Georgiou, and Waters.

About degrees 2 and 3, it could be tested by computer up to about 22 vertices. Would that be useful?

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I have come to suspect that the reconstruction conjecture is "true by accident" and therefore likely to be impossible to prove. In other words, the "chances" (not using the word rigorously) of having two non-isomorphic graphs with isomorphic decks is vanishingly small but there's no particular reason for them not to. –  Gordon Royle Dec 18 '11 at 2:13
Yes, I think a computer search on degrees 2 and 3 would be very useful. (What's the best way to do this? I think I have an old version of Nauty installed on my Mac but it has been a while since I used it -- sorry!) Graphs with degrees 2 & 3 would be (roughly) identified by the vertices of degree 3. "Smooth out" the vertices of degree 2, replacing edge-vertex-edge by a single edge, and one has a cubic graph. Can we reconstruct that cubic graph? –  Ken W. Smith Dec 18 '11 at 13:35
I still have the programs I used to prove the reconstruction conjecture for various small graphs in the early 1990s. I'll dust them off and see if they still run. I'll also see how far I can get with degrees 3 and 4. –  Brendan McKay Dec 18 '11 at 15:01