## The converse of Kelly’s lemma and control-classes in graph reconstruction.(English)Zbl 1086.05051

Let $$G$$ be a graph on $$n$$ vertices. Kelly’s lemma in graph reconstruction states that, from the deck of vertex-deleted subgraphs of $$G$$, the number $$\binom{G}{H}$$ of subgraphs of $$G$$ isomorphic to $$H$$ can be determined for all graphs $$H$$ of order less than $$n$$. The authors here prove the converse of this lemma and some variations. Therefore the reconstruction conjecture can be stated as follows: if all the $$\binom{G}{H}$$, for $$| V(H)| <n$$, are known, then this determines $$G$$ uniquely. This motivates some ways of strenghtening the conjecture. For example, if $$\mathcal K$$ is a class of graphs and $$\binom{G}{H}$$ is given for all graphs $$H$$ in $$\mathcal K$$ and $$| V(H)| <n$$, does this determine $$G$$ uniquely? The authors discuss this and related problems briefly, referring to a later publication. They observe, for example, that if $$\mathcal K$$ is the class of paths and $$G$$ is a tree, then $$G$$ need not be reconstructible this way, but that the question is open when, say, $$\mathcal K$$ is the class of caterpillars or disjoint unions of paths.
They then pose another strengthening of the reconstruction conjecture, namely, the study of the function $$L(n)$$ defined as follows. Let $$L(n)$$ denote the smallest positive integer $$m$$ such that all graphs of order $$n$$ are determined by their induced subgraphs of order $$m$$. The authors give the first eight values of $$L(n)$$ and observe that, assuming that the reconstruction conjecture is true, the plot of $$L(n)$$ against $$n$$ lies below the straight line $$y=x-1$$; but by a result of V. Nýdl [Discrete Math. 108, 373–377 (1992; Zbl 0759.05067)], it does not lie below any straight line passing through the origin and with slope $$q<1$$. (The authors call the function $$L$$ the Ulam-ladder.)

### MSC:

 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)

Zbl 0759.05067
Full Text:

### References:

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