A refinement of Kelly's lemma for graph reconstruction for counting rooted subgraphs

Abstract

Kelly's lemma is a basic result on graph reconstruction. It states that given the deck of a graph $G$ on $n$ vertices, and a graph $F$ on fewer than $n$ vertices, we can count the number of subgraphs of $G$ that are isomorphic to $F$. Moreover, for a given card $G-v$ in the deck, we can count the number of subgraphs of $G$ that are isomorphic to $F$ and that contain $v$. We consider the problem of refining the lemma to count rooted subgraphs such that the root vertex coincides the deleted vertex. We show that such counting is not possible in general, but a multiset of rooted subgraphs of a fixed height $k$ can be counted if $G$ has radius more than $k$. We also prove a similar result for the edge reconstruction problem.