Degree-associated reconstruction number of graphs

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 $drn\left(G\right)$ is the minimum number of dacards that determine $G$. We show that $drn\left(G\right)=2$ for almost all graphs and determine when $drn\left(G\right)=1$. For $k$-regular $n$-vertex graphs, $drn\left(G\right)\le min\{k+2,n-k+1\}$. For vertex-transitive graphs (not complete or edgeless), we show that $drn\left(G\right)\ge 3$, give a sufficient condition for equality, and construct examples with large $drn$. Our most difficult result is that $drn\left(G\right)=2$ for all caterpillars except stars and one 6-vertex example. We conjecture that $drn\left(G\right)\le 2$ for all but finitely many trees.