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Degree-associated reconstruction number of graphs
Authors:Michael D Barrus
Institution:a Department of Mathematics, Black Hills State University, Spearfish, SD 61801, United States
b Department of Mathematics, University of Illinois, Urbana, IL 61801, United States
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)≤min{k+2,nk+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.
Keywords:Reconstruction conjecture  Reconstruction number  Vertex-transitive graph  Tree  Caterpillar
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