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? 
 A: 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?
A: 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.
A: This seems to be a little more recent that 1977:
http://www.akcejournal.org/contents/vol1no1/Vol_1no_1-6.pdf
A: Some recent papers of mine settle some new cases of the graph reconstruction
conjecture.  Recall that a limb of a graph $G$ is a maximal subtree, and the
trunk is $G$ with $L-r$ removed for each limb $L$ with root $r$ (or empty if
is $G$ a tree).  It has long been known that the limbs and trunk are
reconstructible, $G$ is reconstructible if it is a tree, and $G$ is
reconstructible if it has no limbs and the trunk has more than one block.
In "Strong Reconstructibility of the Block-Cutpoint Tree" it is shown that
$G$ is reconstructible if it is not a "single block trunk" graph, i.e.,
its trunk has a single block.
Let $p$ be the number of edges minus the number of nodes.  In "Two Cases
of Reconstruction of Separable Graphs" it is shown that a single block
trunk graph is reconstructible if $p=0$ or $p=1$.
In "Some Results on Reconstructibility of Colored Graphs" the following
are shown.  A graph which is not a single block trunk graph, with a
vertex and edge coloring, is reconstructible.  A block with vertex
colors is reconstructible if $0\leq p\leq 2$.  Edge colored versions of
$K_5$ are not reconstructible, refuting a 1970 conjecture of B. Manvel.
It has yet to be detemined if single block trunk graphs with $p=2$ are
reconstructible.
A: The edge-reconstruction of graphs with degrees k and k-1 has been proved, in 2984 I think, by Ellingham, Myrvold and Hoffman. You can find it in J Graph Theory Vol 11. It is a very clever proof and it's difficulty indicates how much more difficult it is to obtain vertex-reconstruction. Even the vertex-reconstruction of graphs with degrees 2 and 3 is one very tantalising problem because it looks so easy but it is not.
