Degree-associated reconstruction number of graphs

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Abstract

A card of a graph G is a subgraph formed by deleting one vertex. The Reconstruction Conjecture states that each graph with at least three vertices is determined by its multiset of cards. A dacard specifies the degree of the deleted vertex along with the card. The degree-associated reconstruction number drn(G) is the minimum number of dacards that determine G. We show that drn(G)=2 for almost all graphs and determine when drn(G)=1. For k-regular n-vertex graphs, drn(G)≤mink+2,n-k+1. For vertex-transitive graphs (not complete or edgeless), we show that drn(G)<3, give a sufficient condition for equality, and construct examples with large drn. Our most difficult result is that drn(G)=2 for all caterpillars except stars and one 6-vertex example. We conjecture that drn(G)≤2 for all but finitely many trees. © 2010 Elsevier B.V.

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Barrus, M. D., & West, D. B. (2010). Degree-associated reconstruction number of graphs. Discrete Mathematics, 310(20), 2600–2612. https://doi.org/10.1016/j.disc.2010.03.037

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