Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reconstruction of C_4-free graphs from the set of closed neighborhoods and digital convexity

Steffen Borgwardt, MacKenzie Carr, Ce Chen, Wayne Ge, Stephen G. Hartke, Yixuan Huang, Alex Moon

TL;DR

The paper proves that every $C_4$-free graph is strongly reconstructible from both the set of closed neighborhoods $\mathcal{N}[G]$ and the support $\text{supp}(\mathcal{N}[G])$, and that this is equivalent to reconstructibility from the digital convexity $\mathscr{D}(G)$. It provides an alternative proof of a key multiset result, develops the union-basis machinery via $U(\mathcal{N}[G])$ to extract base vertices, and reduces reconstruction to a quotient graph $G'$ that can be reconstructed from $\mathcal{N}[G']$ before inflating back to $G$. The work extends prior results by Lafrance et al. and shows that all graphs of girth at least five (in particular $C_4$-free graphs) are reconstructible from digitally convex sets, connecting convexity concepts to neighborhood-based graph reconstruction. This advances our understanding of when a graph is uniquely determined by compact neighborhood-derived data and provides constructive pathways for reconstruction algorithms within the $C_4$-free class.

Abstract

Fomin, Kratochvíl, Lokshtanov, Mancini, and Telle showed that every $C_{4}$-free graph is reconstructible from the \emph{multiset} of closed neighborhoods. We strengthen their result proving that every $C_{4}$-free graph is reconstructible from the \emph{set} of closed neighborhoods. This extends the work of Lafrance et al.\ by showing that all $C_{4}$-free graphs, and hence all graphs of girth at least five, are reconstructible from their digitally convex sets. A subset $S$ of vertices in a graph $G$ is digitally convex if, for every vertex $v \notin S$, there is a private neighbor of $v$. We establish that reconstruction from digitally convex sets is equivalent to reconstruction from the set of closed neighborhoods.

Reconstruction of C_4-free graphs from the set of closed neighborhoods and digital convexity

TL;DR

The paper proves that every -free graph is strongly reconstructible from both the set of closed neighborhoods and the support , and that this is equivalent to reconstructibility from the digital convexity . It provides an alternative proof of a key multiset result, develops the union-basis machinery via to extract base vertices, and reduces reconstruction to a quotient graph that can be reconstructed from before inflating back to . The work extends prior results by Lafrance et al. and shows that all graphs of girth at least five (in particular -free graphs) are reconstructible from digitally convex sets, connecting convexity concepts to neighborhood-based graph reconstruction. This advances our understanding of when a graph is uniquely determined by compact neighborhood-derived data and provides constructive pathways for reconstruction algorithms within the -free class.

Abstract

Fomin, Kratochvíl, Lokshtanov, Mancini, and Telle showed that every -free graph is reconstructible from the \emph{multiset} of closed neighborhoods. We strengthen their result proving that every -free graph is reconstructible from the \emph{set} of closed neighborhoods. This extends the work of Lafrance et al.\ by showing that all -free graphs, and hence all graphs of girth at least five, are reconstructible from their digitally convex sets. A subset of vertices in a graph is digitally convex if, for every vertex , there is a private neighbor of . We establish that reconstruction from digitally convex sets is equivalent to reconstruction from the set of closed neighborhoods.
Paper Structure (4 sections, 16 theorems, 5 equations, 5 figures)

This paper contains 4 sections, 16 theorems, 5 equations, 5 figures.

Table of Contents

  1. Introduction
  2. An alternative proof of Theorem \ref{['thm: C4 multiset']}
  3. Union of closed neighborhoods
  4. Proof of Theorems \ref{['thm: C4 set']} and \ref{['thm:DC=CN-recon']}

Key Result

Theorem 1.1

If $G$ is a $C_4$-free graph, then $G$ is strongly reconstructible from its closed neighborhoods $\mathcal{N}[G]$.

Figures (5)

  • Figure 1: Two non-isomorphic graphs that have the same set of open neighborhoods.
  • Figure 2: Two non-isomorphic graphs that have the same set of closed neighborhoods.
  • Figure 3: A $C_4$-free graph $G$.
  • Figure 5: $C_4$ with three different labelings.
  • Figure 6: $G$ contains an induced $C_4$.

Theorems & Definitions (25)

  • Theorem 1.1: Fomin, Kratochvíl, Lokshtanov, Mancini, and Telle Fomin
  • Theorem 1.2
  • Theorem 1.3: Lafrance, Oellermann, and Pressey LAFRANCE2017254
  • Proposition 1.4
  • proof
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 2.1
  • Lemma 2.3
  • ...and 15 more