The standard reconstruction conjecture states that a graph is determined by its **deck of vertex-deleted subgraphs**.

Question: Have other decks been investigated, finding out that only vertex-deleted subgraphs can do the job? If so: Which property of vertex-deleted subgraphs makes them exceptional?

I have three candidates in mind, others are conceivable. (For the sake of simplicity I consider only simple connected graphs $G$.)

the

**deck of sub-maximal neighbourhoods**: Let the*sub-maximal neighbourhood of $v$*be the $v$-rooted graph constructed from $G$ by deleting all vertices with maximal distance from $v$.the

**deck of distinguishing neighbourhoods**: Let the*distinguishing neighbourhood of $v$*be the smallest $n$-neighbourhood of $v$ which distinguishes it from all vertices not conjugate to it ($n$-neighbourhood = the $v$-rooted induced subgraph containing all vertices $w$ with distance $d(v,w) \leq n$).the

**deck of crossref-deleted subgraphs**: Let the*crossref-deleted subgraph with respect to $v$*be the $v$-rooted graph constructed from $G$ by deleting all edges between vertices that have the same distance from $v$.

Note that the *vertex-deleted subgraph with respect to $v$* is nothing but the $v$-rooted graph constructed from $G$ by deleting all edges between $v$ and its neighbours.

I am not good in systematically constructing counterexamples, and I do not have very much intuition about general graphs. So, any counterexample to one of the candidates above would be very welcome.

What I *do* know is that a) trees are trivially reconstructible from their deck of crossref-deleted subgraphs, that b) graphs with one node of which the distinguishing neighbourhood is the whole graph are trivially reconstructible from their deck of distinguishing neighbourhoods, and that c) reconstructing (very) small graphs from one of the decks above is fun.