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

Charlotte Knierim, Anders Martinsson

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.

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

Abstract

The Reconstruction Conjecture of Kelly and Ulam states that any graph with vertices can be reconstructed from the multiset of unlabelled subgraphs for all . We refer to as the \emph{deck} of and as the cards of . 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 can be reconstructed from a (potentially adversarially chosen) collection of cards for some . In this paper, we show that the clique count of is reconstructible for all but one size of clique from any cards. We extend this result by showing that for graphs with average degree at most we can reconstruct the -count for all , and that for we can reconstruct the -count for every graph on vertices.
Paper Structure (8 sections, 20 theorems, 39 equations, 1 figure)

This paper contains 8 sections, 20 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of this paper
  3. Proof of Theorem \ref{['thrm:main']}
  4. Tools
  5. Proof of the main result
  6. Case I: $|\mathcal{A}|=\ell$
  7. Case II: $|\mathcal{A}|<\ell$
  8. The case $r+\ell = n$

Key Result

Theorem 1.2

For $n$ sufficiently large and $k\le 0.05\sqrt{n}$, the number of edges $m$ of a graph $G$ on $n$ vertices is reconstructible from any $n-k$ cards.

Figures (1)

  • Figure 1: Illustration of Case $(2)$ of Lemma \ref{['lem:main']}. Given $n-1$ cards that do not uniquely determine $\mathrm{k}_{r}(G)$, there are only two possibilities for which vertex of $G$ could be the removed vertex from the cards in $\mathcal{D}'_\Delta$.

Theorems & Definitions (36)

  • Conjecture 1.1
  • Theorem 1.2: groenland2021size
  • Theorem 1.3: groenland2021reconstructing
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.2: myrvold1992degree
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 26 more