Reconstructibility of the $K_r$-count from $n-1$ cards

TL;DR

This paper investigates the reconstructibility of clique counts in graphs from partial information, showing that most clique counts can be reconstructed from nearly complete subgraph data, advancing understanding of the graph reconstruction problem.

Contribution

It proves that the $K_r$-count is reconstructible from $n-1$ cards for all but one size, and extends results to graphs with bounded average degree and small cliques.

Findings

Reconstructibility of $K_r$-counts from $n-1$ cards for all but one size.

Reconstruction of $K_r$-counts for graphs with average degree at most $3n/8 - O(1)$.

Reconstruction of $K_r$-counts for all $r \,\leq\, \log_2 n$ in any $n$-vertex graph.

Abstract

The Reconstruction Conjecture of Kelly and Ulam states that any graph $G$ with $n\geq 3$ vertices can be reconstructed from the multiset $\mathcal{D}(G)$ of unlabelled subgraphs $G-v$ for all $v\in V(G)$. We refer to $\mathcal{D}(G)$ as the \emph{deck} of $G$ and $G-v\in \mathcal{D}(G)$ as the cards of $G$. This was posed in the 1940s and is still wide open today. In an effort to understand reconstructibility better, a growing collection of research is concerned with understanding what properties of $G$ can be reconstructed from a (potentially adversarially chosen) collection of $k$ cards for some $k< n$. In this paper, we show that the clique count of $G$ is reconstructible for all but one size of clique from any $n-1$ cards. We extend this result by showing that for graphs with average degree at most $3n/8-O(1)$ we can reconstruct the $K_r$-count for all $r$, and that for $r\le \log_2 n$ we can reconstruct the $K_r$-count for every graph on $n$ vertices.