Geodesic
A reduction of the Graph Reconstruction Conjecture
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 3, pp. 529-537

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A graph is said to be reconstructible if it is determined up to isomorphism from the collection of all its one-vertex deleted unlabeled subgraphs. Reconstruction Conjecture (RC) asserts that all graphs on at least three vertices are reconstructible. In this paper, we prove that interval-regular graphs and some new classes of graphs are reconstructible and show that RC is true if and only if all non-geodetic and non-interval-regular blocks G with diam(G) = 2 or diam(G) = diam(Ḡ) = 3 are reconstructible.
Keywords: reconstruction, diameter, geodetic graph, interval-regular graph 
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Monikandan, S.; Balakumar, J. A reduction of the Graph Reconstruction Conjecture. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 3, pp. 529-537. http://geodesic.mathdoc.fr/item/DMGT_2014_34_3_a4/